The document contains 7 questions about finding the volume of solids obtained by rotating regions bounded by curves about axes. Each question gives the bounding curves and axis of rotation, and asks the reader to find the volume of the solid formed.

1. Volumes of
Revolution
Exercises

2. Question 1
Find the volume of the solid bounded by the
two following by revolution around the x­axis.

3. Question 1 Find the volume of the solid bounded by the
two following by revolution around the x­axis.

4. Question 1 Find the volume of the solid bounded by the
two following by revolution around the x­axis.

5. Question 1 Find the volume of the solid bounded by the
two following by revolution around the x­axis.

6. Question 1 Find the volume of the solid bounded by the
two following by revolution around the x­axis.
Intersections

7. Question 1

8. Question 2
The region R is bounded by y = ln(x),
y = 0, x = 1 and x = 2. Find the volume of
the solid obtained by revolving R about the
y­axis.

9. The region R is bounded by y = ln(x),
Question 2 y = 0, x = 1 and x = 2. Find the volume of
the solid obtained by revolving R about the
y­axis.

10. The region R is bounded by y = ln(x),
Question 2 y = 0, x = 1 and x = 2. Find the volume of
the solid obtained by revolving R about the
y­axis.

11. The region R is bounded by y = ln(x),
Question 2 y = 0, x = 1 and x = 2. Find the volume of
the solid obtained by revolving R about the
y­axis.

12. The region R is bounded by y = ln(x),
Question 2 y = 0, x = 1 and x = 2. Find the volume of
the solid obtained by revolving R about the
y­axis.

13. The region R is bounded by y = ln(x),
Question 2 y = 0, x = 1 and x = 2. Find the volume of
the solid obtained by revolving R about the
y­axis.
Answer given on worksheet:

14. Question 3
Find the volume of the solid defined by
revolving the area bounded by y=x^2, y=0
and x=2 about the x­axis.

15. Find the volume of the solid defined by
Question 3 revolving the area bounded by y=x^2, y=0
and x=2 about the x­axis.
Intersection

16. Find the volume of the solid defined by
Question 3 revolving the area bounded by y=x^2, y=0
and x=2 about the x­axis.
Intersection

17. Find the volume of the solid defined by
Question 3 revolving the area bounded by y=x^2, y=0
and x=2 about the x­axis.

18. Find the volume of the solid defined by
Question 3 revolving the area bounded by y=x^2, y=0
and x=2 about the x­axis.

19. Find the volume of the solid defined by
Question 3 revolving the area bounded by y=x^2, y=0
and x=2 about the x­axis.
Answer given on sheet:

20. Question 4
The equations y = sqr(4+x), x=0 and y=0
define the bounds of a region of the plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

21. Question 4 The equations y = sqr(4+x), x=0 and y=0
define the bounds of a region of the plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

22. Question 4 The equations y = sqr(4+x), x=0 and y=0
define the bounds of a region of the plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

23. Question 4 The equations y = sqr(4+x), x=0 and y=0
define the bounds of a region of the plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

24. Question 5
the equations x=1, x=3,y=(1/x) and y=0
define the bounds of a region of a plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

25. Question 5 the equations x=1, x=3,y=(1/x) and y=0
define the bounds of a region of a plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

26. Question 5 the equations x=1, x=3,y=(1/x) and y=0
define the bounds of a region of a plane.
Find the voume of the solid obtained by
rotating the region about the x axis.

27. Question 5 the equations x=1, x=3,y=(1/x) and y=0
define the bounds of a region of a plane.
Find the voume of the solid obtained by
rotating the region about the x axis.
Answer on sheet:

28. Question 6
The equations y=x^2­x and y=0 define
the bounds of a region of a plane. Find
the volume of the solid obtained by
rotating the region about the x­axis.

29. Question 6 The equations y=x^2­x and y=0 define
the bounds of a region of a plane. Find
the volume of the solid obtained by
rotating the region about the x­axis.

30. Question 6 The equations y=x^2­x and y=0 define
the bounds of a region of a plane. Find
the volume of the solid obtained by
rotating the region about the x­axis.

31. Question 6 The equations y=x^2­x and y=0 define
the bounds of a region of a plane. Find
the volume of the solid obtained by
rotating the region about the x­axis.

32. Question 6 The equations y=x^2­x and y=0 define
the bounds of a region of a plane. Find
the volume of the solid obtained by
rotating the region about the x­axis.

33. Question 7
The equations x=­1, x=0, y=(1/(x­1)^3) and
y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

34. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

35. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

36. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

37. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

38. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

39. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.

40. The equations x=­1, x=0, y=(1/(x­1)^3) and
Question 7 y=0 define the bounds of a region of the
plane. Find the volume of the solid obtained
by rotating the region about the x­axis.
Nothing else matters, subtracting negative
infinity makes this infinitely large which
really makes sense since the object just
keeps on going as it never reaches the x
axis.

41. Answer on the sheet... 31pi/160