Volume: The Disc Method

1. Volume: The Disc Method Section 6.2

2. If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disc, which is formed by revolving a rectangle about an axis .

3. Revolving a Function f(x) a b 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?

4. Disks dx 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?

5. If a region of a plane is revolved about a line, the resulting solid is the solid of revolution, and the lineis the axis of revolution. The simplest is a right circular cylinder or disk, a rectangle revolved around the x-axis. The volume is equal to the area of the disk times the width of the disk, V = πr2w. r w

6. The Disc Method Volume of disc = (area of disc)(width of disc)

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

8. w w R R Axis of Revolution

9. Revolve this rectangle about the x-axis. Revolve this function about the x-axis.

10. Revolve this rectangle about the x-axis. Forms a Disk. Revolve this function about the x-axis.

11. Revolving About y-Axis • Also possible to revolve a function about the y-axis • Make a disk or a washer to be horizontal • Consider revolving a parabola about the y-axis • How to represent the radius? • What is the thicknessof the disk?

12. Revolving About y-Axis • Must consider curve asx = f(y) • Radius = f(y) • Slice is dy thick • Volume of the solid rotatedabout y-axis

13. Horizontal Axis of Revolution Volume = V = Vertical Axis of Revolution Volume = V =

14. Disk Method (to find the volume of a solid of revolution) Horizontal Axis of revolution: Vertical Axis of revolution:

15. Disk Method (to find the volume of a solid of revolution) Horizontal Axis of revolution: width radius Vertical Axis of revolution:

16. 1. Find the volume of the solid formed by revolving f(x) over about the x-axis.

17. 1. Find the volume of the solid formed by revolving f(x) over about the x-axis.

18. 2. Find the volume of the solid formed by revolving the region formed by f(x) and g(x) about y = 1.

19. 2. Find the volume of the solid formed by revolving the region formed by f(x) and g(x) about y = 1. Length of Rectangle:

20. 3. Find the volume of the solid formed by revolving the region formed by over about the y-axis.

21. 3. Find the volume of the solid formed by revolving the region formed by over about the y-axis.

22. The disc method can be extended to cover solids of revolution with holes by replacing the representative disc with a representative washer. w R r

23. Washers f(x) g(x) a b Consider the area between two functions rotated about the axis Now we have a hollow solid We will sum the volumes of washers As an integral

24. Volume of washer =

25. How do you find the volume of the figure formed by revolving the shaded area about the x-axis?

26. How do you find the volume of the figure formed by revolving the shaded area about the x-axis? Outside radius Inside radius The volume we want is the difference between the two. Revolving this function creates a solid whose volume is larger than we want. Revolving this function carves out the part we don’t want.

27. Washer Method (for finding the volume of a solid of revolution)

28. Washer Method (for finding the volume of a solid of revolution) Outside radius Inside radius

29. 1. Find the volume of the solid generated by revolving the area enclosed by the two functions about the x- axis.

30. 1. Find the volume of the solid generated by revolving the area enclosed by the two functions about the x- axis. Find intersection points first. Washer Method:

31. 2. Find the volume of the solid formed by revolving the area enclosed by the given functions about the y- axis.

32. 2. Find the volume of the solid formed by revolving the area enclosed by the given functions about the y- axis. This region is not always created by the same two functions. The change occurs at y = 1. We need to use two integrals to find the volume. For y in [0,1] we use disk method. For y in [1,2] we use washer method. Since we’re revolving about the y-axis, each radius must be in terms of y. (distance from the parabola to the x-axis) (distance from the parabola to the y-axis)

34. Volume of Revolution - X Find the volume of revolution about the x-axis of f(x) = sin(x) + 2 from x = 0 to x = 2.

35. Volume of Revolution - X Find the volume of revolution about the x-axis of f(x) = sin(x) + 2 from x = 0 to x = 2 Use the TI to integrate this one! Did you get 88.826 cubic units?

36. Volumes of Solids with Known Cross Sections • For cross sections of area A(x) taken perpendicular to the x-axis, Volume = • For cross sections of area A(y) taken perpendicular to the y-axis, Volume =

37. Volume of Revolution - X Let’s look at the cross section or slice. What is it? It’s a circle. What is the radius? f(x). What is the area? A = r2 A =  (f(x))2

38. Volume of Revolution - X How wide (thick) is the disc? dx The volume of the disk is V =  (f(x))2dx How do we add up all the disks from x = 1 to x = 4?

39. Example 1 Find the volume of the solid whose base is the region in the xy-plane bounded by the given curves and whose cross-sections perpendicular to the x-axis are (a) squares; (b) semicircles; and (c) equilateral triangles.

40. Example 1(a) Square Cross Sections • My Advice??? • Draw the 2-D picture and imagine the cross-sectional shape coming out in the third dimension! • Try and think of how to write the area of these cross sections using the given function. • Set up the integral and integrate. Think: SKATE BOARD RAMP!!!

41. Example 1(b) Semicircular Cross Sections Think: CORNICOPIA!!!

42. Example 1(c) Equilateral Triangular Cross Sections Think: PYRAMID, kinda??

43. Example 2 Find the volume of the solid whose base is the region in the xy-plane bounded by the given curves and whose cross-sections perpendicular to the x-axis are (a) squares; (b) semicircles; and (c) equilateral triangles.

44. Example 2 (a) Square Cross Sections Think: PYRAMID!!!

45. Example 2 (b) Semicircular Cross Sections Think: CORNICOPIA AGAIN!!!

46. Example 2 (c) Equilateral Triangular Cross Sections Think: PYRAMID!!!

47. Example 2 (c) Equilateral Triangular Cross Sections Think: PYRAMID!!!

48. Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by a b Finding the volume of a solid with known cross-section is a 3-step process:

49. Theorem: The volume of a solid with cross-section of area A(x) that is perpendicular to the x-axis is given by ***This circle is the base of a 3-D figure coming out of the screen. S a b ***This rectangle is a side of a geometric figure (a cross-section of the whole). Finding the volume of a solid with known cross-section is a 3-step process: Step #1: Find the length (S) of the rectangle used to create the base of the figure. Step #2: Find the area A(x) of each cross-section (in terms of this rectangle). Step #3: Integrate the area function from the lower to the upper bound. Volume of a solid with cross-sections of area A(y) and perpendicular to the y-axis: d S c

50. Find the volume of the solid whose base is bounded by the circle with cross-sections perpendicular to the x-axis. These cross-sections are a. squares Step #1: Find S. Step #2: Find A(x). Step #3: Integrate the area function.