There are good hints and notes in the answers to the following, but struggle first before peeking at those!


 Imogene Brenda Hancock
 10 months ago
 Views:
Transcription
1 Integration Worksheet  Using the Definite Integral Show all work on your paper as described in class. Video links are included throughout for instruction on how to do the various types of problems. Important: Work the problems to match everything that was shown in the videos. For eample: Suppose a video shows ways to do a problem, (such as algebraically, graphically, and numerically), then your work should show these ways also. That is, each video is a model for the work I want to see on your paper. ESSAY. 1R: Read and take notes on Section 8.1 1)Finding Areas by Slicing Watch and take notes on these videos to see the methods and terminology I want you to use: There are good hints and notes in the answers to the following, but struggle first before peeking at those! (Worksheet items  9 below have selected eercises from Section 8.1) ) (a) Write a Riemann sum approimating the area of the region in the figure, using vertical strips as shown. (b) Evaluate the corresponding integral. Follow the details as shown in the above videos. ) (a) Write a Riemann sum approimating the area of the region in the figure, using vertical strips as shown. (b) Evaluate the corresponding integral. Follow the details as shown in the above videos. 1
2 4) (a) Write a Riemann sum approimating the area of the region in the figure, using horizontal strips as shown. (b) Evaluate the corresponding integral. Follow the details as shown in the above videos. ) (a) Write a Riemann sum approimating the area of the region in the figure, using horizontal strips as shown. (b) Evaluate the corresponding integral. Follow the details as shown in the above videos.
3 )Do,. Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral eactly.
4 7)Do 7,8. Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral eactly. 4
5 8)Do 9, 11. Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral eactly.
6 9) Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral eactly. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the area of the shaded region. 1) f() = +  y g() = (, 18) 1 (, ) (4, 4)  A) 4 1 B) 81 1 C) 97 1 D) 78 1
7 11) y = y y = A) B) C) 9 D) 1) y 1 1 y = y = 4 A) B) 7 1 C) 7 1 D) 1 7
8 1) y = +  y =  4 y (, 4) A) 11 B) 9 C) 19 D) 8 14) 4 y y =  4 y = A) B) 4 C) D) 18 1) 1 y y = y = sin(π) A) 8 B) 4 π C) π D) 4 8
9 Find the area enclosed by the given curves. 1) y = , y =  4 A) 4 B) 1 C) 7 D) 17) y =, y = A) 1 B) 1 C) 1 D) ) Find the area of the region in the first quadrant bounded by the line y = 8, the line = 1, the curve y = 1, and the ais. A) 4 B) C) D) 4 19) Find the area of the region in the first quadrant bounded on the left by the yais, below by the line y = 1, above left by y = + 4, and above right by y = A) 7 B) 9 C) 1 D) 9 4 Volumes of solids  VERY IMPORTANT! ESSAY. )Watch and take notes on this video. (It shows an eample of finding the volume of a special solid created by revolving a region about a line). I promised at the beginning of the course that you would be able to do this, and you will, very soon! 1) Go to and manipulate this applet by working with the sliders to see how a solid (a cylinder in this case) is cut into "slabs". The idea is that each slab volume would be calculated and then all these would be added up to get the total volume of the cylinder. Copy the figure onto your paper showing the cuts. )Watch and take notes on these videos. They will help you with the net section of problems. (Worksheet items  7 below contain selected eercises from Section 8.1) 9
10 ) Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. 4)Do 14, 1. Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. 1
11 )Do 1, 17, 18 Write a Riemann sum and then a definite integral representing the volume of the region, using the slice shown. 11
12 ) A dam has a rectangular base 14 meters long and 1 meters wide. Its crosssection is shown in the figure. (The Grand Coulee Dam in Washington state is roughly this size.) By slicing horizontally, set up and evaluate a definite integral giving the volume of material used to build this dam. 7) The figure shows a solid with both rectangular and triangular cross sections. (a) Slice the solid parallel to the triangular faces. Sketch one slice and calculate its volume in terms of, the distance of the slice from one end. Then write and evaluate an integral giving the volume of the solid. (b) Repeat part (a) for horizontal slices. Instead of, use h, the distance of a slice from the top. Volumes of Solids of Revolution 8R: Read and take notes on Section 8. 8) Watch and take notes on these video illustrations (draw the pictures) that show how regions are rotated about a line to create a solid, and then cutting up the solid into disks, cuts, slabs, rings, slabs with holes etc. to find the volume of the solid
13 9) Go to and move the angle slider down to rotate the region which then creates the solid of revolution. Copy the graphs to your paper. ) Go to and copy the region and resulting D rotations from eample 1,,, 4 onto your paper. Take notes. 1) Go to and click on the gifs to show the animation of the cuts of the solids. Copy the pictures to your paper. )Watch and take notes on these videos. They will help you with this net section of problems: (Worksheet items  7, 9 below contain selected problems from Section 8.) ) a) The region in in the figure is rotated around the ais. Using the strip shown, write an integral giving the volume. b) Evaluate the integral. 4) (a) The region in the figure is rotated around the yais. Write an integral giving the volume. (b) Evaluate the integral. 1
14 )Do 71 odd: The region described is rotated around the ais. Find the volume. (in case you are wondering about #1, hyperbolic cosine (cosh) is covered in Calculus ) )Do, 7: Set up, but do not evaluate, an integral that represents the volume obtained when the region in the first quadrant is rotated about the given ais. 7) Do 7, 9: Sketch the solid obtained by rotating each region around the indicated ais. Using the sketch, show how to approimate the volume of the solid by a Riemann sum, and hence find the volume. 8) Go to and copy down the pictures for each of the solids of known cross section. These are NOT solid of revolution, they are solids created by known (square, semicircle, triangle) cross sections. These types of D images are hard to imagine so this demonstration helps you see them. Watch and take notes on this video: 14
15 9)Do 41, 4, 4. These concern the region bounded by y =, y = 1, and the yais, for. Find the volume of the following solids. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the volume of the described solid. 4) The solid lies between planes perpendicular to the ais at = and =. The cross sections perpendicular to the ais between these planes are squares whose bases run from the parabola y =  to the parabola y =. A) B) 48 C) 18 D) 4 41) The solid lies between planes perpendicular to the ais at =  and =. The cross sections perpendicular to the ais between these planes are squares whose bases run from the semicircle y = to the semicircle y = 4 . A) 4 B) C) 18 D) 1 4) The base of the solid is the disk + y. The cross sections by planes perpendicular to the yais between y =  and y = are isosceles right triangles with one leg in the disk. A) 1 B) C) 1 D) 4) The solid lies between planes perpendicular to the ais at =  and =. The cross sections perpendicular to the ais are semicircles whose diameters run from y = to y = 4 . A) 1 4 π B) π C) π D) 8 π 1
16 44) The solid lies between planes perpendicular to the ais at =  4 and = 4. The cross sections perpendicular to the ais are circular disks whose diameters run from the parabola y = to the parabola y = . A) π B) π C) π D) 1 1 π 4) The base of a solid is the region between the curve y = cos and the ais from = to = π/. The cross sections perpendicular to the ais are squares with bases running from the ais to the curve. A) π B) 9 π C) π D) 9 4 π Find the volume of the solid generated by revolving the shaded region about the given ais. 4) About the ais y y = A) 1π B) 4 4 π C) π D) π 47) About the ais y y = A) 1 π B) π C) π D) 18π 1
17 48) About the yais y = y/ A) 98π B) 1π C) 18π D) 84π 49) About the yais y 4 y = A) π B) π C) π D) π ) About the yais 4 1 y = y 1 4 A) 7 π B) 18 4 π C) π D) 18π 17
18 1) About the ais y y = A) 4 1 π B) 8 4 π C) π D) 1 1 π ) About the ais 4 y y = sin π A) 9 π  9π B) 9 π C) 9 π  π D) 9 π + 9π Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the ais. ) y =, y =, =, = A) 1π B) π C) 8 π D) 4 π 4) y =, y =, =, = A) 777 π B) 7π C) 4π D) 1944π ) y = +, y =, =, = 1 A) π B) π C) 4π D) π 18
19 ) y = 1, y =, = 1, = 8 A) 7 1 π B) πln 8 C) 8 π D) 7 8 π 7) y =, y = 1, = A) 144 π B) π C) π D) π Find the volume of the solid generated by revolving the region about the given line. 8) The region bounded above by the line y = 1, below by the curve y = 1 , and on the right by the line = 4, about the line y = 1 A) π B) π C) π D) 1 π 9) The region in the second quadrant bounded above by the curve y = 4 , below by the ais, and on the right by the yais, about the line = 1 A) 1 π B) π C) π D) 8π Solve the problem. ) The disk (  ) + y 4 is revolved about the yais to generate a torus. Find its volume. (Hint: 4  y dy = π, since it is the area of a semicircle of radius.)  A) 4π B) 48π C) 1π D) 4π 1) The hemispherical bowl of radius contains water to a depth 1. Find the volume of water in the bowl. A) π B) π C) π D) 88π ) A water tank is formed by revolving the curve y = 4 about the yais. Find the volume of water in the tank as a function of the water depth, y. A) V(y) = π y / B) V(y) = π y / π C) V(y) = y 1/ D) V(y) = π 9 y 9 ) A right circular cylinder is obtained by revolving the region enclosed by the line = r, the ais, and the line y = h, about the yais. Find the volume of the cylinder. A) πrh B) πrh C) πrh D) πrh 19
20 4) Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius. A) π B) 8 1 π C) π D) π ESSAY. The Shell Method, another method for finding the volume of a solid of revolution. ) Watch and take notes on this video illustrating the Shell Method, another method for finding the volume of a solid of revolution. Especially pay attention to the applet he shows at 9 seconds into the video: AND, watch this gif and take notes: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated ais. ) About the yais y y = A) π B) 4π C) 8 π D) 4 π
21 7) About the ais y 4 1 y = A) π B) π C) 1 π D) 1π 8) About the yais y 4 1 y =sin() 1.8 A) 1π B) π C) 9π D) π Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the yais. 9) y =, y = 4 +, for A) π B) 19π C) 9π D) 4π 7) y = 7, y =  7, = 1 A) 7 π B) 1 1 π C) π D) π 1 71) y =, y =, = 1, = 4 A) 4 8 π B) π C) 4π D) π 1
22 7) y = 4e, y =, =, = 1 A) 811 e π B) 4(e  1) π C) 411 e π D) e π Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the ais. 7) = y, =  y, y = 1 A) 11 π B) π C) 8π D) π 74) y =, y =, y =  A) π B) π C) 7π D) 4 π Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. 7) y = 4 , y = 4, = ; revolve about the line y = 4 A) 4 π B) 1 1 π C) 8 π D) π 7) y =, y =, = ; revolve about the line =  A) 14 4 π B) π C)  π D) ESSAY. Finding the length of a curve (Arclength) 77) Watch and take notes on this video: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the length of the curve. 78) y = / from = to = 4 A) 9 4 B) C) 7 D) 18 79) y = from = 1 to = A) 1 1 B) 79 C) 1 D) 17
23 Set up an integral for the length of the curve. 8) y = 4, 1 1 A) d B) d C) d D) d Work 81R: Read and take notes on Section 8. JUST THE WORK PART!) ESSAY. 81)Write these notes on your paper: In physics the word work has a technical meaning which is different from its everyday meaning. Physicists say that if a constant force, F, is applied to some object to move it a distance, d, then the force has done work on the object. The force must be parallel to the motion (in the same or the opposite direction). We make the following definition: Work done = Force Distance or W = F d We'll see two types of WORK problems here: 1) The work needed to lift objects up to a location, or to pump liquid up and out of a container, ) The work needed to build structures from the ground up. 8)Watch this video about WORK : The back story is that Maynard G. Crebs is a "beatnik" (in the 19's) and therefore stereotypically doesn't like the idea of "work": You may feel the same about WORK after you do the following problems! 8) An anchor weighing 1 lb in water is attached to a chain weighing lb/ft in water. Find the work done to haul the anchor and chain to the surface of the water from a depth of ft. 84) A construction worker is standing on the roof of a ft. above the ground. A lbs tool bag is attached to a rope weighing 4 lbs per foot. Find the work needed to lift the bag from the ground up to the roof. 8) A rectangular water tank has length ft., width 1 ft., and depth 1 ft. If the tank is full, how much work does it take to pump all the water out (up and out or the top)? Water weighs.4 lbs/ft 8) A water tank is in the form of a right circular cylinder with height ft and radius ft. If the tank is half full of water, find the work required to pump all of it over the top rim. Water weighs.4 lbs/ft 87) Suppose the tank in the previous problem is full of water. Find the work required to pump all of it to a point 8 ft above the top of the tank. Water weighs.4 lbs/ft
24 88) Water in a cylinder of height 1 ft and radius 4 ft is to be pumped out. Water weighs.4 lbs/ft. Find the work required if (a) The tank is full of water and the water is to pumped over the top of the tank. (b) The tank is full of water and the water must be pumped to a height ft above the top of the tank. (c) The depth of water in the tank is 8 ft and the water must be pumped over the top of the tank. 89) A radius 1 ft, height 1 ft. cone monument (as shown below) is to be constructed with stones weighing lbs/ft. How much work is required? Hint: You want to first find out how much a slab cut up at a y height weighs and how much work it will take to lift it from the ground up to that y location. So you need the radius of the slab cut. Set up a ratio to get this in terms of y. 9) It is reported that the Great Pyramid of Egypt was built in years. If the stone making up the pyramid has a density of pounds per cubic foot, find the total amount of work done in building the pyramid. The pyramid is 41 feet high and has a square base 7 feet by 7 feet. Hint: Set up a ratio to get the length of the slab cut, s, up at a h height location. (I used s and h here since the answer key does.) 91) In the previous problem about the Great Pyramid of Egypt, you calculated the total work done in building the pyramid. Now estimate the total number of workers needed. Assume every laborer worked 1 hours a day, days a year, for years. Assume that a typical worker lifted ten pound blocks a distance of 4 feet every hour. Doing some calculations and your answer to the previous problem, how many workers were needed? 9) A cylindrical garbage can of depth ft and radius 1 ft fills with rainwater up to a depth of ft. How much work would be done in pumping the water up to the top edge of the can? (Water weighs.4 lb/ft.) Solve the problem. 9) A ft long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of lbs/ft, How much work is required to wind the entire chain onto the cylinder using the winch? 4
25 94) A small pool has the shape of a bo with a base that measures ft by 19 ft and a depth of ft. How much work is required to pump the water out of the pool when it is full? (density of water =.4 lbs ft ) 9) A cylindrical water tank has height 1 ft and radius ft (see figure). If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? (density of water =.4 lbs ft ) 1 ft ft 9) A spherical water tank with an inner radius of 4 ft has its lowest point ft above the ground. It is filled by a pipe that feeds the tank at its lowest point (see figure). Neglecting the volume of the inflow pipe, how much work is required to fill the tank if it is initially empty? Hint: Think of the tank after it is full and consider how much work it took to lift each slab of water in the sphere from the ground up to its position. 4 ft ft (density of water =.4 lbs ft )
26 97) A water trough has a semicircular cross section with a radius of ft and a length of ft (see figure). How much work is required to pump water out of the trough when it is full? (density of water =.4 lbs ft ) ft ft 98) A glass has circular cross sections that taper (linearly) from a radius of 8 in. at the top of the glass to a radius of 7 in. at the bottom. The glass is 14 in. high and full of lemonade. How much work is required to drink all the lemonade through a straw if your mouth is in. above the top of the glass? Assume the density of lemonade equals the density of water, (density of water =.4 lbs lbs =.1 ft in. ) THE END!!!
27 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL 1) ) a) Area (This is the shorthand sum notation we want. We don't bother with the i counter, we : to just want to set up a sum that continually reminds us we are adding up slices ) And, notice the Riemann sum is an approimation so we put " " or " ", whereas the integral is the eact area because the limit has been taken, so we put the "=" sign. b) Area = d = 9 (The last step is to set up the integral and evaluate it. Verify by geometry, you can (as in this one) and verify your byhand algebra simplifications by running the Integral program. ) a) Area ( + ) b) Area = ( + )d = : to 4) a) Area ( y/)δy b) Area = ( y/)dy = 9.You always should get the same answer using y: to vertical or horizontal cuts (although one direction may be harder than the other). ) a) We must solve for in terms of y to see up at a y location what the left value and right value of the strip is. y =  + =>  + y = => = ± 9y (after simplifying from the quadratic formula) => the length of the strip up at a y location is: ( + 9y )( 9y ) = 9y so Area 9 9y y b) Area = 9y dy = This is the same area answer as was obtained in y: to 9 the previous problem with the same figure, it just turned out to be a much more difficult direction to do the cuts. You always should get the same answer using vertical or horizontal cuts (although one direction may be harder than the other). ) ) Area, Area = d= 1 ) To see what the y height value is at an location, set up the : to similar triangle ratio: y = => y = (1/). As moves right to left (as the picture instructs us to do) we obtain the following sum for the area of the figure: Area (1/), Area = : to (1/) d= 9 7
28 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL 7) 7) Label the strip width a. Then set up a similar triangle ratio: a h = => a = (h). So up at an h height location the width of the strip is (h). So Area (h) h. Lets check the etremes before going h: to on: Down on the ground h = so the strip should be wide there: () =, check! Up at the top h = so the strip should be wide there: () =, check! So Area = (h) d = 1 8) This one was shown in a video: The answer is 9π 8) 9) Up at a y height location the width of the strip is the value solved for in terms of y: = 1y so Area 1y y, Area = 1 1y dy = (/)π (eact value, but for this one just run y: to 1 the Integral program, I'll take it! ) 11) You must solve each equation for in terms of y to find the width of the strip up at a y height location, so Area 1 (yy) y, Area = (yy)dy = 1/ y: to 1 9) The height of the strip at an location is () (  4) which simplifies to so Area ( ), Area = ( )d = 4 : to 1) C 11) C 1) C 1) C 14) B 1) D 1) D 17) C 18) A 19) A ) 1) ) 9 ) Each slab is a cylinder with volume π(), so Volume 4π cm, Volume = 4π d = : to 9 π 8
29 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL 4) 14) We must find the radius of the slab at an location. Set up ratio: r = => r = (1/), Volume of slab = π( 9 ) π Volume cm, Volume = 9 : to 1) answer: Volume = π 9 d = 8π cm ) 1) Each slab cut is a rectangular solid. We must find the width of the rectangle side up at a y location. Width = 7 49y, so slab volume = 49y (1) y, Volume 49y y m, Volume = 49y dy = y: to 7 4π m (For this one I would ecept you running the Integral program to get a decimal approimation.) 17) answer: 18) Up at a y location we find the width of the slab : a y = => a = y, so the volume of the slab = (y)(y) y, Volume (y) y, Volume = (y) dy = (y) dy = (y) evaluated from to = y: to  8 = 8 m (notice the trick of turning the subtraction around when the quantity is an even power.) Notice also that the volume of a pyramid is (1/) (area base) (height) = (1/)(4)() = 8 m so the calculus answer checks out. ) answer: 9
30 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL 7) answer: 8) 9) ) 1) ) ) Draw the picture, then draw a representative cut (a disk). At an location this disk has radius so the volume of the disk is π(). Develop the sum and then the integral from there (: to ) 4)
31 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL ) answers: ) answers: 7) answers: 8) 9) answers: 1
32 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL 4) B 41) C 4) A 4) A 44) C 4) D 4) C 47) B 48) D 49) B ) B 1) C ) A ) C 4) A ) C ) D 7) B 8) D 9) C ) B 1) B ) B ) D 4) C ) ) C 7) A 8) B 9) D 7) B 71) A 7) C 7) A 74) B 7) D 7) B 77) 78) D 79) A 8) D 81) 8) 8) see video 84) ()() + 4(y) dy = = 18 ftlbs. 8) see the video
33 Answer Key Testname: INTEGRATION WORKSHEET  USING THE DEFINITE INTEGRAL 8) see the video 87) Work π y.4 (8y), Work = π(.4)(8y) dy= 418. ftlbs y: to 88) (a) 1,88 ftlbs (b) 1, ftlbs (c) 1, ftlbs a 89) 1y = 1 1, get a in terms of y. Then, the weight of the slab = π( 1 1 (1y)) y ft lbs ft So the work needed to lift the slab from the ground up to the y location is: π( 1 1 (1y)) y ft lbs ft y ft. = π( 1 1 (1y)) y y ftlbs 1 Work = π( 1 1 (1y)) y dy = ftlbs 9) Answer: 91) answer: 9) ftlb 9) 1 ftlbs 94) 4744 ftlbs 9) 881 ftlbs 9) ftlbs 97) 1.8 ftlbs 98) 94. ftlbs
Contents. 12 Applications of the Definite Integral Area Solids of Revolution Surface Area...
Contents 12 Applications of the Definite Integral 179 12.1 Area.................................................. 179 12.2 Solids of Revolution......................................... 181 12.3 Surface
More informationApplications of Integration Day 1
Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under
More informationx = y + 2, and the line
WS 8.: Areas between Curves Name Date Period Worksheet 8. Areas between Curves Show all work on a separate sheet of paper. No calculator unless stated. Multiple Choice. Let R be the region in the first
More information0 3 x dx W = 2 x2 = = inlb x dx W = = 20. = = 160 inlb.
Friday, October 9, 215 p. 83: 5, 7, 11, 12, 15, 17, 19, 2, 21, 25, 26 Problem 5 Problem. A force of 5 pounds compresses a 15inch spring a total of 3 inches. How much work is done in compressing the spring
More informationChapter 5 Applications of Integration
MA111 Application of Integration Asst.Prof.Dr.Supranee Lisawadi 1 Chapter 5 Applications of Integration Section 5.1 Area Between Two Curves In this section we use integrals to find areas of regions that
More informationWeek #15  Word Problems & Differential Equations Section 8.1
Week #15  Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by HughesHallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by
More information6.1 Area between curves
6. Area between curves 6.. Area between the curve and the ais Definition 6.. Let f() be continuous on [a, b]. The area of the region between the graph of f() and the ais is A = b a f()d Let f() be continuous
More informationName Calculus AP Chapter 7 Outline M. C.
Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and nonnegative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)
More information7.3 Volumes Calculus
7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are
More informationDavis Buenger Math Solutions September 8, 2015
Davis Buenger Math 117 6.7 Solutions September 8, 15 1. Find the mass{ of the thin bar with density 1 x function ρ(x) 1 + x < x. Solution: As indicated by the box above, to find the mass of a linear object
More informationTest Two Review Cal 2
Name: Class: Date: ID: A Test Two Review Cal 2 Short Answer 1. Set up the definite integral that gives the area of the region bounded by the graph of y 1 x 2 2x 1 and y 2 2x.. Find the area of the region
More information1. AREA BETWEEN the CURVES
1 The area between two curves The Volume of the Solid of revolution (by slicing) 1. AREA BETWEEN the CURVES da = {( outer function ) ( inner )} dx function b b A = da = [y 1 (x) y (x)]dx a a d d A = da
More informationPROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving
More informationWork the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.
CALCULUS WORKSHEET ON 7. Work the following on notebook paper. Find the area bounded by the given graphs. Show all work. Do not use your calculator.. f x x and g x x on x Find the area bounded by the given
More informationW i f(x i ) x. i=1. f(x i ) x = i=1
Work Force If an object is moving in a straight line with position function s(t), then the force F on the object at time t is the product of the mass of the object times its acceleration. F = m d2 s dt
More informationDefinition: Let S be a solid with crosssectional area A(x) perpendicular to the xaxis at each point x [a, b]. The volume of S is V = A(x)dx.
Section 7.: Volume Let S be a solid and suppose that the area of the crosssection of S in the plane P x perpendicular to the xaxis passing through x is A(x) for a x b. Consider slicing the solid into
More informationM1120 Class 5. Dan Barbasch. September 4, Dan Barbasch () M1120 Class 5 September 4, / 16
M1120 Class 5 Dan Barbasch September 4, 2011 Dan Barbasch () M1120 Class 5 September 4, 2011 1 / 16 Course Website http://www.math.cornell.edu/ web1120/index.html Dan Barbasch () M1120 Class 5 September
More informationChapter 8. Chapter 8 Opener. Section 8.1. Big Ideas Math Green WorkedOut Solutions. Try It Yourself (p. 353) Number of cubes: 7
Chapter 8 Opener Try It Yourself (p. 5). The figure is a square.. The figure is a rectangle.. The figure is a trapezoid. g. Number cubes: 7. a. Sample answer: 4. There are 5 6 0 unit cubes in each layer.
More informationApplications of Integration to Geometry
Applications of Integration to Geometry Volumes of Revolution We can create a solid having circular crosssections by revolving regions in the plane along a line, giving a solid of revolution. Note that
More informationDIFFERENTIATION OPTIMIZATION PROBLEMS
DIFFERENTIATION OPTIMIZATION PROBLEMS Question 1 (***) 4cm 64cm figure 1 figure An open bo is to be made out of a rectangular piece of card measuring 64 cm by 4 cm. Figure 1 shows how a square of side
More informationCHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often
7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationIntroduction. Rectangular Volume. Module #5: GeometryVolume Bus 130 1:15
Module #5: Geometryolume Bus 0 :5 Introduction In this module, we are going to review the formulas for calculating volumes, rectangular, cylindrical, and irregular shaped. To set the stage, let s suppose
More information3D Geometry: Chapter Questions
3D Geometry: Chapter Questions 1. What are the similarities and differences between prisms and pyramids? 2. How are polyhedrons named? 3. How do you find the crosssection of 3Dimensional figures? 4.
More informationCHAPTER 29 VOLUMES AND SURFACE AREAS OF COMMON SOLIDS
CHAPTER 9 VOLUMES AND SURFACE AREAS OF COMMON EXERCISE 14 Page 9 SOLIDS 1. Change a volume of 1 00 000 cm to cubic metres. 1m = 10 cm or 1cm = 10 6m 6 Hence, 1 00 000 cm = 1 00 000 10 6m = 1. m. Change
More informationCHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.
TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has
More information1. Kyle stacks 30 sheets of paper as shown to the right. Each sheet weighs about 5 g. How can you find the weight of the whole stack?
Prisms and Cylinders Answer Key Vocabulary: cylinder, height (of a cylinder or prism), prism, volume Prior Knowledge Questions (Do these BEFORE using the Gizmo.) [Note: The purpose of these questions is
More informationApplications of the Integral
Chapter 6 Applications of the Integral Evaluating integrals can be tedious and difficult. Mathematica makes this work relatively easy. For example, when computing the area of a region the corresponding
More informationCONNECT: Volume, Surface Area
CONNECT: Volume, Surface Area 1. VOLUMES OF SOLIDS A solid is a threedimensional (3D) object, that is, it has length, width and height. One of these dimensions is sometimes called thickness or depth.
More informationWork. Example. If a block is pushed by a constant force of 200 lb. Through a distance of 20 ft., then the total work done is 4000 ftlbs. 20 ft.
Work Definition. If a constant force F is exerted on an object, and as a result the object moves a distance d in the direction of the force, then the work done is Fd. Example. If a block is pushed by a
More information121 Representations of ThreeDimensional Figures
Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 121 Representations of ThreeDimensional Figures Use isometric dot paper to sketch each prism. 1. triangular
More informationGeometry Chapter 12. Volume. Surface Area. Similar shapes ratio area & volume
Geometry Chapter 12 Volume Surface Area Similar shapes ratio area & volume Date Due Section Topics Assignment Written Exercises 12.1 Prisms Altitude Lateral Faces/Edges Right vs. Oblique Cylinders 12.3
More informationGeometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources:
Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students
More information126 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. SOLUTION: ANSWER: 1017.
Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 3. sphere: area of great circle = 36π yd 2 We know that the area of a great circle is r.. Find 1. Now find the surface area.
More informationArea of Parallelograms, Triangles, and Trapezoids (pages 314 318)
Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base
More information1. A plane passes through the apex (top point) of a cone and then through its base. What geometric figure will be formed from this intersection?
Student Name: Teacher: Date: District: Description: MiamiDade County Public Schools Geometry Topic 7: 3Dimensional Shapes 1. A plane passes through the apex (top point) of a cone and then through its
More information17.1 Cross Sections and Solids of Rotation
Name Class Date 17.1 Cross Sections and Solids of Rotation Essential Question: What tools can you use to visualize solid figures accurately? Explore G.10.A Identify the shapes of twodimensional crosssections
More information124 Volumes of Prisms and Cylinders. Find the volume of each prism.
Find the volume of each prism. 3. the oblique rectangular prism shown at the right 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. If two solids have
More informationThe small increase in x is. and the corresponding increase in y is. Therefore
Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that
More information128 Congruent and Similar Solids
Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. 3. Two similar cylinders have radii of 15 inches and 6 inches. What is the ratio
More informationTeacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.
Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 91.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles
More informationLESSON SUMMARY. Measuring Shapes
LESSON SUMMARY CXC CSEC MATHEMATICS UNIT SIX: Measurement Lesson 11 Measuring Shapes Textbook: Mathematics, A Complete Course by Raymond Toolsie, Volume 1 (Some helpful exercises and page numbers are given
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationHeight. Right Prism. Dates, assignments, and quizzes subject to change without advance notice.
Name: Period GL UNIT 11: SOLIDS I can define, identify and illustrate the following terms: Face Isometric View Net Edge Polyhedron Volume Vertex Cylinder Hemisphere Cone Cross section Height Pyramid Prism
More informationArea is a measure of how much space is occupied by a figure. 1cm 1cm
Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number
More informationGeometry Second Semester Final Exam Review
Name: Class: Date: ID: A Geometry Second Semester Final Exam Review 1. Mr. Jones has taken a survey of college students and found that 1 out of 6 students are liberal arts majors. If a college has 7000
More informationWebAssign Lesson 33 Applications (Homework)
WebAssign Lesson 33 Applications (Homework) Current Score : / 27 Due : Tuesday, July 15 2014 11:00 AM MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /3 points A
More informationA mountain climber is about to haul up a 50m length of hanging rope. How much work will it take if the rope weighs N/m?
WORK, FLUID PRESSURES AND FORCES WORK When we push a chair a distance along a straightline with constant force, we can determine the work done by the force on the chair by using the formula W = Fd. Unfortunately,
More informationGeometry Work Samples Practice
1. If ABCD is a square and ABE is an equilateral triangle, then what is the measure of angle AFE? (H.1G.5) 2. Square ABCD has a circular arc of radius 2 with the center at A and another circular arc with
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationSurface Area and Volume
UNIT 7 Surface Area and Volume Managers of companies that produce food products must decide how to package their goods, which is not as simple as you might think. Many factors play into the decision of
More informationArea of Parallelograms (pages 546 549)
A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular
More informationFree Response Questions Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom
Free Response Questions 1969005 Compiled by Kaye Autrey for facetoface student instruction in the AP Calculus classroom 1 AP Calculus FreeResponse Questions 1969 AB 1 Consider the following functions
More informationMathematical Modeling and Optimization Problems Answers
MATH& 141 Mathematical Modeling and Optimization Problems Answers 1. You are designing a rectangular poster which is to have 150 square inches of tet with inch margins at the top and bottom of the poster
More information14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Objectives. Iterated Integrals. Iterated Integrals
14 Multiple Integration 14.1 Iterated Integrals and Area in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Evaluate an iterated
More informationStudent Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)
Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.
More informationSURFACE AREA AND VOLUME
SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has
More informationThe graphs of f and g intersect at (0, 0) and one other point. Find that point: f(y) = g(y) y 2 4y 2y 2 6y = = 2y y 2. 2y(y 3) = 0
. Compute the area between the curves x y 4y and x y y. Let f(y) y 4y y(y 4). f(y) when y or y 4. Let g(y) y y y( y). g(y) when y or y. x 3 y? The graphs of f and g intersect at (, ) and one other point.
More information12 Surface Area and Volume
12 Surface Area and Volume 12.1 ThreeDimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids
More informationGCSE Exam Questions on Volume Question 1. (AQA June 2003 Intermediate Paper 2 Calculator OK) A large carton contains 4 litres of orange juice.
Question 1. (AQA June 2003 Intermediate Paper 2 Calculator OK) A large carton contains 4 litres of orange juice. Cylindrical glasses of height 10 cm and radius 3 cm are to be filled from the carton. How
More informationSandia High School Geometry Second Semester FINAL EXAM. Mark the letter to the single, correct (or most accurate) answer to each problem.
Sandia High School Geometry Second Semester FINL EXM Name: Mark the letter to the single, correct (or most accurate) answer to each problem.. What is the value of in the triangle on the right?.. 6. D.
More informationEngineering Math II Spring 2015 Solutions for Class Activity #2
Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the xaxis. Then find
More informationFS Geometry EOC Review
MAFS.912.GC.1.1 Dilation of a Line: Center on the Line In the figure, points A, B, and C are collinear. http://www.cpalms.org/public/previewresource/preview/72776 1. Graph the images of points A, B, and
More informationMAC Calculus II Spring Homework #2 SOLUTIONS
MAC 232593Calculus II Spring 23 Homework #2 SOLUTIONS Note. Beginning already with this homework, sloppiness costs points. Missing dx (or du, depending on the variable of integration), missing parentheses,
More informationNorthfield Mount Hermon School Advanced Placement Calculus. Problems in the Use of the Definite Integral
Northfield Mount Hermon School Advanced Placement Calculus Problems in the Use of the Definite Integral Dick Peller Winter 999 PROBLEM. Oil is leaking from a tanker at the rate of R(t) = e .t gallons
More informationSection 2.4: Applications and Writing Functions
CHAPTER 2 Polynomial and Rational Functions Section 2.4: Applications and Writing Functions Setting up Functions to Solve Applied Problems Maximum or Minimum Value of a Quadratic Function Setting up Functions
More information27.2. Multiple Integrals over Nonrectangular Regions. Introduction. Prerequisites. Learning Outcomes
Multiple Integrals over Nonrectangular Regions 7. Introduction In the previous Section we saw how to evaluate double integrals over simple rectangular regions. We now see how to etend this to nonrectangular
More information128 Congruent and Similar Solids
Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. Ratio of radii: Ratio of heights: The ratios of the corresponding measures are
More information111. Space Figures and Cross Sections. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
111 Space Figures and Cross Sections Vocabulary Review Complete each statement with the correct word from the list. edge edges vertex vertices 1. A(n) 9 is a segment that is formed by the intersections
More informationLesson 11: The Volume Formula of a Pyramid and Cone
Classwork Exploratory Challenge Use the provided manipulatives to aid you in answering the questions below. a. i. What is the formula to find the area of a triangle? ii. Explain why the formula works.
More informationSection 6.4: Work. We illustrate with an example.
Section 6.4: Work 1. Work Performed by a Constant Force Riemann sums are useful in many aspects of mathematics and the physical sciences than just geometry. To illustrate one of its major uses in physics,
More informationName: Date: Geometry Solid Geometry. Name: Teacher: Pd:
Name: Date: Geometry 20122013 Solid Geometry Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the Volume of Prisms and Cylinders Pgs: 17 HW: Pgs: 810 DAY 2: SWBAT: Calculate the Volume of
More informationchp12review Name: Class: Date: 1. Find the number of vertices, faces, and edges for the figure below.
Name: Class: Date: chp12review 1. Find the number of vertices, faces, and edges for the figure below. a. 6 vertices, 6 faces, 10 edges b. 5 vertices, 6 faces, 10 edges c. 7 vertices, 7 faces, 11 edges
More informationTotal Time: 90 mins Practise Exercise no. 1 Total no. Of Qs: 51
Mensuration (3D) Total Time: 90 mins Practise Exercise no. 1 Total no. Of Qs: 51 Q1. A wooden box of dimensions 8m x 7m x 6m is to carry rectangular boxes of dimensions 8cm x 7cm x 6cm. The maximum number
More information4 More Applications of Definite Integrals: Volumes, arclength and other matters
4 More Applications of Definite Integrals: Volumes, arclength and other matters Volumes of surfaces of revolution 4. Find the volume of a cone whose height h is equal to its base radius r, by using the
More informationTERMINOLOGY Area: the two dimensional space inside the boundary of a flat object. It is measured in square units.
SESSION 14: MEASUREMENT KEY CONCEPTS: Surface Area of right prisms, cylinders, spheres, right pyramids and right cones Volume of right prisms, cylinders, spheres, right pyramids and right cones the effect
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationAP Calculus Testbank (Chapter 7) (Mr. Surowski)
AP Calculus Testbank (Chapter 7) (Mr. Surowski) Part I. MultipleChoice Questions. Suppose that a function = f() is given with f() for 4. If the area bounded b the curves = f(), =, =, and = 4 is revolved
More information*1. Understand the concept of a constant number like pi. Know the formula for the circumference and area of a circle.
Students: 1. Students deepen their understanding of measurement of plane and solid shapes and use this understanding to solve problems. *1. Understand the concept of a constant number like pi. Know the
More informationSection 2.1 Rectangular Coordinate Systems
P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is
More informationSection P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities
Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.
More informationPage 2 6. A semicircular trough is shown. If x = 5 m and y = 1 m, how many cubic meters of fluid will this trough hold? 7. In the diagram, the base of
Accerlerated Geometry Name In Class Practice: Area/Volume Per/Sec. Date (1.) Show work for credit. (2.) Leave solution in exact form unless otherwise stated. (3.) Provide the correct UNITS on all solutions.
More informationGeometry Notes VOLUME AND SURFACE AREA
Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate
More informationGeo  CH10 Practice Test
Geo  H10 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. lassify the figure. Name the vertices, edges, and base. a. triangular pyramid vertices:,,,,
More information( ) where W is work, f(x) is force as a function of distance, and x is distance.
Work by Integration 1. Finding the work required to stretch a spring 2. Finding the work required to wind a wire around a drum 3. Finding the work required to pump liquid from a tank 4. Finding the work
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationHigher Mathematics Homework A
Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(3,1) and B(5,3) 2. Find the equation of the tangent to the circle
More informationSection 63 DoubleAngle and HalfAngle Identities
63 DoubleAngle and HalfAngle Identities 47 Section 63 DoubleAngle and HalfAngle Identities DoubleAngle Identities HalfAngle Identities This section develops another important set of identities
More informationGrade 11 Essential Mathematics Unit 6: Measurement and Geometry
Grade 11 Essential Mathematics Unit 6: INTRODUCTION When people first began to take measurements, they would use parts of the hands and arms. For example, a digit was the width of a thumb. This kind of
More informationAssignment 7  Solutions Math 209 Fall 2008
Assignment 7  Solutions Math 9 Fall 8. Sec. 5., eercise 8. Use polar coordinates to evaluate the double integral + y da, R where R is the region that lies to the left of the yais between the circles
More informationLesson 25: Volume of Right Prisms
Lesson 25 Lesson 25: Volume of Right Prisms Classwork Opening Exercise Take your copy of the following figure, and cut it into four pieces along the dotted lines (the vertical line is the altitude, and
More information122 Surface Areas of Prisms and Cylinders. 1. Find the lateral area of the prism. SOLUTION: ANSWER: in 2
1. Find the lateral area of the prism. 3. The base of the prism is a right triangle with the legs 8 ft and 6 ft long. Use the Pythagorean Theorem to find the length of the hypotenuse of the base. 112.5
More informationSolids. Objective A: Volume of a Solids
Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular
More informationPlatonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples:
Solid Geometry Solid Geometry is the geometry of threedimensional space, the kind of space we live in. Three Dimensions It is called threedimensional or 3D because there are three dimensions: width,
More informationVolumes by Cylindrical Shell
Volumes by Cylindrical Shell Problem: Let f be continuous and nonnegative on [a, b], and let R be the region that is bounded above by y = f(x), below by the x  axis, and on the sides by the lines x =
More informationSOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Solid Shapes Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier SOLID SHAPES Version: 2.1 Date: 10112015 Mathematics Revision Guides Solid
More informationOverview Mathematical Practices Congruence
Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason
More informationConverting measurements. 10 mm 1 cm 1000 mm 1 m 100 cm 1 m 1000 m 1 km. Change 3 km to cm. 3 km m (as 1 km 1000 m) cm
Unit 2: Number 2 Converting measurements Converting lengths Remember 10 mm 1 cm 1000 mm 1 m 100 cm 1 m 1000 m 1 km Unit 2: Number 2 Eample 1 Change 3 km to cm. 3 km 3 1000 m (as 1 km 1000 m) 3 1000 100
More information