Exercises

Identify each integral as either type I or type II and evaluate:

1. ó õ 1 0  ó õ 1 0  ( x+y) dydx 2. ó õ 2 0  ó õ 3 1  x2y  dydx 3. ó õ 2 0  ó õ 3 0  xy  dxdy 4. ó õ 1 0  ó õ 3 0  dydx 5. ó õ 1 0  ó õ x 0  ( x2+y2) dydx 6. ó õ p 0  ó õ sin( x) 0  dydx 7. ó õ p 0  ó õ p 0  cos( x) dydx 8. ó õ p 0  ó õ x 0  sin( y) dydx 9. ó õ p/4 0  ó õ sec( x) tan( x) 0  dydx 10. ó õ 2p 0  ó õ sin( x) 0  ydydx 11. ó õ p 0  ó õ x 0  sin( x) dydx 12. ó õ 1 0  ó õ y 0  ex+ydxdy 13. ó õ p 0  ó õ exp( x) 0  xdydx 14. ó õ 1 0  ó õ y 0  sin( y2) dxdy 15. ó õ 2 0  ó õ y 0  ln( y2+1) dxdy 16. ó õ 3 0  ó õ 1 x  eydxdy 17. ó õ 2 1  ó õ x2 0   x x2+y2 dydx 18. ó õ 2 1  ó õ x 0   1 x2+y2 dydx

Sketch the region R and determine its type. Then find the volume of the solid under z = f( x,y)  and over the given region.

19. f( x,y) = x2+y2 20. f( x,y) = 3 R: y in [0, 1] R: x in [0, 2] x = 0, x = 1 y = 0, y = 4 21. f( x,y) = 3x+2y 22. f( x,y) = 6x+y R: x in [0, 1] R: x in [2,3] y = 0, y = x2 y = 0, y = ex 23. f( x,y) = xy 24. f( x,y) = y2 R: y in [0, 1] R: y in [0, p/2] x = -y, x = y x = 0, x = sin( y) 25. f( x,y) = ex+y 26. f( x,y) = 9-x2-y2 R: y in [0,1] R: x in [1,3] x = 0, x = 1-y y = x, y = x2

The following regions are unbounded. Sketch the region R and determine its type. Then find the volume of the solid under z = f( x,y)  and over the given region.

27. f( x,y) =  1 x2y2 28. f(x,y) =  1 x2+y2 R: x in ( 1,¥) , y in ( 1,¥) R: x in (0, ¥) , y in (0, ¥) 29. f( x,y) = x-2e-y 30. f(x,y) = 1 R: x in ( 1,¥) R: x in (0,¥) y = 0, y = x-2 y = x-e-x, y = x+e-x

31. A regular cone with a height h and a base with radius R is positioned so that its axis is horizontal. Find the area A( x) of a vertical cross-section of the cone perpendicular to the axis as a function of x in [ 0,h] .

What is the volume of a regular cone with height h and a base with radius R?

32. A hemisphere with radius R is positioned so that its axis is horizontal. Find the area A( x) of a vertical cross-section of the cone perpendicular to the axis as a function of x in [0,R] .

What is the volume of a hemisphere with radius R?

33. A regular pyramid has height h and a square base with each side a length s. It is positioned as shown in the figure below:

Find the area A( x) of a cross-section at x. What is the volume of the pyramid?

34. The Great Pyramid is 481^¢ tall and has a square base which is 756^¢ wide on each side.

What is the volume of the Great Pyramid? (hint: see problem 33).

35. Explain why the area of a type I region can be written in the form

A = ó õ b a  ó õ h( x) g( x)   dydx

36. Explain why the area of a type II region can be written in the form

A = ó õ d c  ó õ q( y) p( y)   dxdy

37. Explain why if a,b,c, and d are all constant, then

ó õ b a  ó õ d c  f( x,y)dydx = ó õ d c  ó õ b a  f( x,y) dxdy

when both iterated integrals exist.

38. Show that if a,b,c, and d are constant, then

ó õ b a  ó õ d c  f( x) g( y) dydx = é ë ó õ b a  f( x) dx ù û é ë ó õ d c  g( y)dy ù û

39. Use properties of the integral to show that

ó õ b a  ó õ q( x) p( x)   [ f(x,y) +g( x,y) ] dy dx = ó õ b a  ó õ q( x) p(x)   f( x,y)dy dx+ ó õ b a  ó õ q( x) p( x)   g(x,y) dy dx

40. Use properties of the integral to show that

ó õ b a  ó õ q( x) p( x)   [ f(x,y) +g( x,y) ] dy dx = ó õ b a  ó õ q( x) p(x)   f( x,y)dy dx+ ó õ b a  ó õ q( x) p( x)   g(x,y) dy dx

41. Show that if f is differentiable on ( a,b) , then for all c in ( a,b) we have 

f( c) ( b-a) + ó õ b a  f( x)dx = ó õ b a  ó õ x c  f' ( u) dudx

42. Show that if f is differentiable and if f( 0) = 0, then

ó õ b a  f( x) dx = ó õ b a  ó õ 1 0  f' (ux) dudx