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

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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. 1-R: 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 y-ais, 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 cross-section 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 8-R: 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 y-ais. Write an integral giving the volume. (b) Evaluate the integral. 1

14 )Do 7-1 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 y-ais, 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 y-ais 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 y-ais y = y/ A) 98π B) 1π C) 18π D) 84π 49) About the y-ais y 4 y = A) π B) π C) π D) π ) About the y-ais 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 y-ais, about the line = 1 A) 1 π B) π C) π D) 8π Solve the problem. ) The disk ( - ) + y 4 is revolved about the y-ais 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 y-ais. 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 y-ais. 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 y-ais 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 y-ais 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 y-ais. 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) 81-1 e π B) 4(e - 1) π C) 41-1 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 81-R: 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 short-hand 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 by-hand 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 = => = ± 9-y (after simplifying from the quadratic formula) => the length of the strip up at a y location is: ( + 9-y )-(- 9-y ) = 9-y so Area 9 9-y y b) Area = 9-y 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: = 1-y so Area 1-y y, Area = 1 1-y 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 (y-y) y, Area = (y-y)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 49-y, so slab volume = 49-y (1) y, Volume 49-y y m, Volume = 49-y 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 ft-lbs. 8) see the video

33 Answer Key Testname: INTEGRATION WORKSHEET - USING THE DEFINITE INTEGRAL 8) see the video 87) Work π y.4 (8-y), Work = π(.4)(8-y) dy= 418. ft-lbs y: to 88) (a) 1,88 ft-lbs (b) 1, ft-lbs (c) 1, ft-lbs a 89) 1-y = 1 1, get a in terms of y. Then, the weight of the slab = π( 1 1 (1-y)) y ft lbs ft So the work needed to lift the slab from the ground up to the y location is: π( 1 1 (1-y)) y ft lbs ft y ft. = π( 1 1 (1-y)) y y ft-lbs 1 Work = π( 1 1 (1-y)) y dy = ft-lbs 9) Answer: 91) answer: 9) ft-lb 9) 1 ft-lbs 94) 4744 ft-lbs 9) 881 ft-lbs 9) ft-lbs 97) 1.8 ft-lbs 98) 94. ft-lbs

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