1. Calculus April 11Volume: the Disk Method

2. Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 < x <  ) about the x-axis.

3. Calculus April 11 Volume: the Disk Method Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 < x <  ) about the x-axis. First, the graph of the function:

4. Calculus April 11 Volume: the Disk Method Find the volume of the solid formed by revolving the region bounded by the graph of and the x-axis (0 < x <  ) about the x-axis. First, the graph of the function: Now, a visual of the solid of revolution:

5. Focus on the original graph for a moment, cut into rectangles.

6. The width of each rectangle is, and the height of each rectangle is

7. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h.

8. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

9. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

10. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

11. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

12. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

13. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

14. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

15. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is

16. Focus on the original graph for a moment, cut into rectangles. The width of each rectangle is, and the height of each rectangle is When we revolve about the x-axis, the rectangle creates a cylinder, and we know the volume of a cylinder is  r 2 h. The radius of the cylinder is and the height of the cylinder is Of course the real answer is  cubic units, but the back of the text book will just say .

17. Revolving about a line that is not an axis.

18. f(x) = 2 – x 2, g(x) = 1, revolve about the line y = 1.

19. Revolving about a line that is not an axis. f(x) = 2 – x 2, g(x) = 1, revolve about the line y = 1. Where did the boundary from – 1 to 1 come from?

20. Revolving about a line that is not an axis. f(x) = 2 – x 2, g(x) = 1, revolve about the line y = 1. Where did the boundary from – 1 to 1 come from?

21. Revolving about a line that is not an axis. The radius is not f(x). The radius is f(x) – g(x).

22. Revolving about a line that is not an axis. The radius is not f(x). The radius is f(x) – g(x). 2 – x 2 – 1 = 1 – x 2

23. Revolving about a line that is not an axis. The radius is not f(x). The radius is f(x) – g(x). 2 – x 2 – 1 = 1 – x 2

24. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals?

25. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals? y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis.

26. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals? y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. Now, the outer radius of the cylinder is easy, the radius is 1. The bottom half of the cylinder is solid, so it is just like we have done using the disk method.

27. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals? y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. Now, the outer radius of the cylinder is easy, the radius is 1. The bottom half of the cylinder is solid, so it is just like we have done using the disk method. Now, the top half of the cylinder has a hollow region. The outer radius is still just 1, but the inner radius is, uh, what is the inner radius?

28. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals? y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. Let’s find the outer radius:

29. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals? y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. Let’s find the outer radius:

30. What about hollow regions? What about revolving around the y-axis? What if we need two different integrals? y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. Let’s find the outer radius: So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

31. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

32. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

33. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

34. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

35. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

36. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

37. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

38. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

39. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

40. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.

41. y = x 2 + 1, y = 0, x = 0, x = 1, revolve about the y-axis. So we have disks formed by the rectangular slices from 0 to 1, and washers formed by the rectangular slices from 1 to 2.