Applications of Integration Day 1

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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 the curve y = 1 x from x = 1 to x = 5.

2 Example 3 Find the area bounded by y = sec 2 x, x = 0, x = π 4, y = 0 Example 4 Find the area bounded by y = ex, x = 0, x = ln 4, y = 0

3 Example 5 Find the finite area bounded by the curve y = x3 2x 2 3x and the x-axis. Example 6 Find the finite area bounded by the two graphs y = 2 x and x + y = 3.

4 Example 7 Find the area lying between y = 6x x2 and y = x2 2x. Example 8 Example 9 Determine the area of the region Determine the area of the enclosed by y = e x, y = e x, x = ln 4 region bounded by y = e x, y = 1, y = 2, x = 3

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6 Applications of Integration Day 2 More on Area Between Curves and using the Calculator Example 1 Example 2 Find the area between Find the area enclosed by y = sec 2 x and y = sin x y = 2 x 2 and y = x from x = 0 to x = π 4 Example 3 Find the area enclosed by y = 2cos x and y = x2 1

7 Example 4 Find the area bounded by y = x and the x-axis and the line y = x 2 There exists another way to determine the area above take a horizontal slice.

8 Example 5 Find the area enclosed by y = x3 and x = y 2 2 Example 6 Find the area bounded by x = y 1 ( )( y 3) and the y-axis

9 Summary Concept I Area Region A Region B Region C Region D General Comments Notice that a and b are on the x-axis c and d are on the y-axis When using formulas involving you must solve for for example: When using formulas involving you must solve for for example: Look at the region and decide which way to slice vertical slices horizontal slices Remember if using horizontal slices, write relations in the form If finding total area, sometimes it is necessary to break up your integrals, for example

10 Applications of Integration Day 3 Volumes of Solids by Plane Slices Consider the volume of a cylinder Consider a bounded region and rotating that region around the x-axis. Take a slice and consider its volume. Take the sum of the volumes of each slice. What do you get? Volume formula:

11 Example 1 The region bounded by the curve and the x-axis over the interval [1,4] is rotated about the x-axis to generate a solid. Find the volume of the solid. Find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis Example 3 Example 4

12 Find the volume of the solid generated by revolving the region bounded by the lines and curves about the y-axis Example 5 Example 6 Example 7 The region bounded by the curve x-axis to generate a solid. Find the volume of the solid. is rotated about the

13 Applications of Integration Day 4 Volume formulas for rotation around the axes (based on cross-sectional areas): Disk Washer f(x) around the x-axis region between f(x) and g(x) around x- axis f(y) around the y-axis region between f(y) and g(y) around y-axis More Volumes using the Washer Method Example 1 (1985 exam) The region bounded by the curves y = ex and y = e x is rotated about the x- axis to generate a solid. Find the volume of the solid.

14 Example 2 Find the volume of the solid generated by revolving the region bounded y = x 2 and y = 2x about the y-axis Rotating Regions about lines parallel to the axes. First idea think of distances in terms of x and y values below and pick a point (x, y). Consider distances from the point to the lines y=3 and y= 1. Then consider distances from point to the lines x = 2 and x = 2 Graph y = x2

15 Example 3 The region bounded by the curve y = 2 x2 and y = 1 is rotated about the line y = 1 to generate a solid. Find the volume of the solid. Example 4 5 Example The region bounded by y = x2, x = 0, x = 2 Take the same region in Eg 4 is rotated about the line y = 1 to generate rotate it about the line y = 4 a solid. Find the volume of the solid. to generate a solid. Find the volume

16 Example 5 The region bounded by y = x 2 + 2, y = 1 2 x +1, x = 0, x =1 is rotated about the line y = 3 to generate a solid. Find the volume of the solid. Example 6!!!!!!!! Example 7!!!! The region bounded by y = x2, x = 0, x = 2!! Same region in Eg. 6 is rotated is rotated about the line x = 3 to generate generate a solid. Find the volume of the solid. about the line x = 3 to solid find the volume.

17 Example 8 The region bounded by y = following lines. Find each volume. x, y = 3, x = 0 is rotated about each of the a) x-axis b) y-axis c) the line y = 3 d) the line x = 9 Additional Questions: *use your graphing calculator to help you find the region and intersection points when necessary. 1. Find the area of the region bounded by a) the graph of x = ( y 1) ( y 4) and the y-axis b) the graphs of y = e0.2x and y = cos x between x = 0 and x = 5 c) the graphs of y = x3 4x and y = 3x2 4x 4 (finite region) 2. Let and a) Find coordinates of intersection and sketch the graph b) Find the area bounded by the graphs 3. Find the area of the region bounded by the graphs of and (draw the region)

18 4 A = ( y 1) ( y 4)dy = 4.5 Solutions: 1. a) 1 5 A = ( cos x e 0.2x )dx = 9.55 b) 0 2 A = (( x 3 4x) ( 3x 2 4x 4) )dx = 6.75 c) 1 2. a) = 1.177, b) A = (cosx x 2 +1) dx = A = (x 3 x +1 e x 2 ) dx + (e x 2 x 3 + x 1) dx =

19 Applications of Integration Day 5 First idea: Draw on the axes below. Pick a point on the curve and label it: P (x, y). a) Draw the line y = 1 and find the distance d from the point P to the line b) Draw the line y = 4 and find the distance d from the point P to the line c) Draw the line x = 1 and find the distance d from the point P to the line d) Draw the line x = 3 and find the distance d from the point P to the line Second idea: Review area formulas Circle Semicircle Square Isosceles Right triangle Equilateral triangle

20 Third idea: Consider the solids we have been working with up to this point. Suppose we sliced the solid and pulled out a cross section what would it look like? Look at the volume formula how could this formula be viewed in terms of the cross-sectional area Consider another type of solid. This solid is not formed by rotation of a region about a line. Considering a region as its base, then describing its crosssectional shape forms a solid with a known cross section. How could we write the general formula for the volume of such a shape? Can you imagine a solid having a circular base and each cross-section perpendicular to the x-axis is a square. Make a solid that looks like this. Draw what it must look like. Find its volume.

21 Example 1 Find the volume of the solid where the base is the circle and a) the cross section perpendicular to the x-axis is an isosceles right triangle with one leg in the plane of the base. b) the cross section perpendicular to the x-axis is an equilateral triangle with one leg in the plane of the base.

22 Example 2 Find the volume of the solid where the base is the region bounded by and the x-axis between x = 0 and x = π a) the cross section perpendicular to the x-axis is a square b) the cross section perpendicular to the x-axis is a semicircle

23 Example 3 Find the volume of the solid where the base is the region bounded by, and the line x = ln 2. a) the cross section perpendicular to the x-axis is a square b) the cross section perpendicular to the x-axis is a semicircle There is another way to describe a solid instead of taking a base describe it solely in terms of its cross sectional area.

24 Example 5 Find the volume of an object that lies within [2, 5] and its cross section at each point is a square of length x. Example 6 Find the volume of an object that lies between x = 0 and x = π/2 and its cross section is a square of length cos x.

25 Concept I!Area Calculus 12 AP Unit V Review of Big Ideas Region A Region B Region C Region D General Comments! Notice that a and b are on the x-axis c and d are on the y-axis! When using formulas involving you must solve for for example:! When using formulas involving you must solve for for example:! Look at the region and decide which way to slice!! vertical slices!! horizontal slices! Remember if using horizontal slices, write relations in the form! If finding total volume, sometimes it is necessary to break up your integrals, for example (See Assignments V-1 and V-2 for good questions on area to practice)

26 Concept II! Volume A! Volumes of Revolution General Comments! These are based on circular slices therefore all of the formulas involve! Try to visualize the slice, then rotate it around the line specified!! - then decide if it is a disk!! - or if it is a washer! When revolving around the x-axis take vertical slices! When revolving around the y-axis take horizontal slices Region A Region B Region C Region D When revolving Region A When revolving Region B When revolving Region C When revolving Region D around x-axis visualize the around y-axis visualize the around x-axis visualize the around y-axis visualize the disk disk washer washer Revolving Regions Around lines Parallel to the Axes When revolving around lines parallel to the x-axis and y-axis draw a diagram to determine the value of R and r in the formula for the washer. It is often helpful to think of a point on the curve in the form and to use the distances x and y in your formulas for R and r, then translate them using the function.!!!!!! For example, in the above diagram if the region being rotated is under the curve between x = 0 and x = 2!! - rotate region about the line y = 3! R = 3 + y r = 3

27 !! - rotate region about the line y = 5! R = 5 r = 5 y! In both cases, substitute these values into the formula: B! Volumes of Known Cross Section General Comments! These are based on slices of different shapes, usually squares, right triangles, equilateral triangles or! semicircles therefore you must know the following area formulas square! right triangle!! equilateral triangle semicircle! Try to draw the slice, remove it from the diagram, then decide on the length of s or r.! For example, consider a solid with a base given by the region bounded by the curve, the x-axis! and the lines x = 0 and x = 1.5 If the known cross sections are squares then: Take a vertical slice Since Therefore The important thing in any volume question is to be able to visualize the cross-section by taking a slice. (See Assignments V-3, V-4, and Additional questions for good questions on volume to practice ) Also do Calculus AP Volume Worksheet Assignment V-5 on the next page.

28 Calculus AP Volume Worksheet Assignment V-5 A. In questions 1 to 5, consider the region bounded by, the x-axis, and the lines x = 0 and x = 1. Find the volume of the following solids. (Draw a diagram for each question). 1. The solid obtained by rotating the region about the x-axis. 2. The solid obtained by rotating the region about the line y = The solid obtained by rotating the region about the line y = The solid whose base is the given region and whose cross-sections perpendicular to the x-axis are squares. 5. The solid whose base is the given region and whose cross-sections perpendicular to the x-axis are semicircles. B. For questions 6 to 7, write out the integral that you would use to calculate the volume. Then, use your calculator to find the numerical integral. 6. The region bounded by, rotated about the line x = The region bounded by, rotated about the line x = 2. C. Find the volume of the solid described in questions 8 to 13. Write out a formula for the cross-sectional area, the write the integral you would use to calculate the volume, then use your calculator to find the numerical integral. 8. The cross section is a square with side length x; the object lies between x = 0 and x = The cross section is a square with side length, the object lies between x = 0 and x = π. 10. The base is the circle, the cross section perpendicular to the x-axis is a square. 11. The base is the circle, the cross section perpendicular to the x-axis is an equilateral triangle. 12. The base is the circle, the cross section perpendicular to the x-axis is an isosceles right triangle with one leg in the plane of the base. 13. The base is the circle, the cross section perpendicular to the x-axis is a semicircle whose diameter is in the plane of the base.

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