Applications of Integration

1. Applications of Integration • In this chapter we explore some of the applications of the • definite integral by using it for • Computing the area between curves • Computing the volumes of solids • Computing the work done by a varying force • Computing average value of a function • The common theme is the following general method, which is similar to the one we used to find areas under curves: • We break up a Q quantity into a large number of small parts. We next approximate each small part by a quantity of the form and thus approximate Q by a Riemann sum. Then we take the limit and express Q as an integral Finally we evaluate the integral using the Fundamental Theorem of Calculus or the Midpoint Rule.

2. Area Between Two Curves http://www.math.tamu.edu/AppliedCalc/Classes/Riemann/Diff.html

3. Example

4. Example

5. Example

6. We could split the area into several sections, use subtraction and figure it out, but there is an easier way. How can we find the area between these two curves?

7. Consider a very thin vertical strip. The length of the strip is: or Since the width of the strip is a very small change in x, we could call it dx.

8. Since the strip is a long thin rectangle, the area of the strip is: If we add all the strips, we get:

10. The formula for the area between curves is: We will use this so much, that you won’t need to “memorize” the formula!

13. Some regions are best treated by regarding x as a function of y

14. If we try vertical strips, we have to integrate in two parts: We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y.

15. We can find the same area using a horizontal strip. Since the width of the strip is dy, we find the length of the strip by solving for x in terms of y. length of strip width of strip

16. 1 General Strategy for Area Between Curves: Sketch the curves. Decide on vertical or horizontal strips. (Pick whichever is easier to write formulas for the length of the strip, and/or whichever will let you integrate fewer times.) 2 3 Write an expression for the area of the strip. (If the width is dx, the length must be in terms of x. If the width is dy, the length must be in terms of y. 4 Find the limits of integration. (If using dx, the limits are x values; if using dy, the limits are y values.) 5 Integrate to find area. p

18. 3 3 3 0 h 3 Find the volume of the pyramid: Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. This correlates with the formula: s dh

19. 1 Method of Slicing: Sketch the solid and a typical cross section. Find a formula for V(x). (Note that I used V(x) instead of A(x).) 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.

20. y x If we let h equal the height of the slice then the volume of the slice is: h 45o x A 45o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge shape several ways, but the simplest cross section is a rectangle. Since the wedge is cut at a 45o angle: Since

21. y x Even though we started with a cylinder, p does not enter the calculation!

22. Cavalieri’s Theorem: Two solids with equal altitudes and identical parallel cross sections have the same volume. Identical Cross Sections p

27. Suppose I start with this curve. My boss at the ACME Rocket Company has assigned me to build a nose cone in this shape. So I put a piece of wood in a lathe and turn it to a shape to match the curve.

28. The volume of each flat cylinder (disk) is: How could we find the volume of the cone? One way would be to cut it into a series of thin slices (flat cylinders) and add their volumes. In this case: r= the y value of the function thickness = a small change in x =dx

29. The volume of each flat cylinder (disk) is: If we add the volumes, we get:

30. This application of the method of slicing is called the disk method. The shape of the slice is a disk, so we use the formula for the area of a circle to find the volume of the disk. If the shape is rotated about the x-axis, then the formula is: A shape rotated about the y-axis would be: Since we will be using the disk method to rotate shapes about other lines besides the x-axis, we will not have this formula on the formula quizzes.

31. Revolving a Function • Consider a function f(x) on the interval [a, b] • Now consider revolvingthat segment of curve about the x axis • What kind of functions generated these solids of revolution? f(x) a b

32. dx Disks f(x) • We seek ways of usingintegrals to determine thevolume of these solids • Consider a disk which is a slice of the solid • What is the radius • What is the thickness • What then, is its volume?

33. Disks • To find the volume of the whole solid we sum thevolumes of the disks • Shown as a definite integral f(x) a b

34. The region between the curve , and the y-axis is revolved about the y-axis. Find the volume. y x The radius is the x value of the function . We use a horizontal disk. The thickness is dy. volume of disk

36. The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

37. The volume of the washer is: The region bounded by and is revolved about the y-axis. Find the volume. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”. outer radius inner radius

38. The washer method formula is: This application of the method of slicing is called the washer method. The shape of the slice is a circle with a hole in it, so we subtract the area of the inner circle from the area of the outer circle. Like the disk method, this formula will not be on the formula quizzes. I want you to understand the formula.

39. r R If the same region is rotated about the line x=2: The outer radius is: The inner radius is: p

40. Find the volume of the region bounded by , , and revolved about the y-axis. We can use the washer method if we split it into two parts: cylinder inner radius outer radius thickness of slice Japanese Spider Crab Georgia Aquarium, Atlanta

47. The solids in Examples 1–5 are all called solids of revolution because they are obtained by revolving a region about a line. In general, we calculate the volume of a solid of revolution by using the basic defining formula We find the cross-sectional area in one of the following ways:

49. Here is another way we could approach this problem: cross section If we take a vertical slice and revolve it about the y-axis we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object.

50. cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.