Exercises:

Find the volume of the solid between the graphs of the given functions over the region bounded by the given curves in the xy-plane:

1. f( x,y) = xy, g( x,y) = 0 2. f(x,y) = x+2y, g( x,y) = 0 x = 0, x = 1, y = 0, y = 1 x = 1, x = 2, y = 0, y = 6 3. f( x,y) = x2+y2, g( x,y) = 0 4. f( x,y) = x3+y2, g( x,y) = 0 y = 0, y = 1, x = y, x = 1 y = 1, y = 2, x = y, x = y2 5. f( x,y) = x+y, g( x,y) = x2+y2 6. f( x,y) = xy, g( x,y) = 4 x = 0, x = 1, y = 0, y = 1 y = 0, y = 1, x = y, x = 1 7. f( x,y) = sin( x) , g( x,y) = 1, 8. f( x,y) = cos( x2) , g( x,y) = 1, x = 0, x = p, y = 0, y = x x = 0, x = p, y = 0, y = x

Evaluate the iterated integral by changing it from type I to type II or vice versa:

9. ó õ 1 0  ó õ 1 x  cos( py2) dydx 10. ó õ 1 0  ó õ 1 y  2ysin( px3) dxdy 11. ó õ p 0  ó õ p x   sin( y) y dydx 12. ó õ 1 -1  ó õ 1 | y|   sin(x2y3) dxdy 13. ó õ 1 0  ó õ p/2 sin-1( x)   xcsc(y) dydx 14. ó õ 2 0  ó õ 4 x2  ex/Öy dydx 15. ó õ 4 1  ó õ 2 Öy   1 x+y dxdy 16. ó õ 2 0  ó õ 4-x2 0   xe2y 4-y dydx

Evaluate using Fubini's theorem.

17. ó õ 1 0  ó õ 2p 0  xsin( y)  dydx 18. ó õ 1 -1  ó õ 3 0  xsin( y2) dxdy 19. ó õ 1 0  ó õ 3 0  ex+y dydx 20. ó õ 1 -1  ó õ 3 0  sinh( xy) dxdy 21. ó õ p -p  ó õ p 0   sin( x2y) dxdy 22. ó õ p/4 -p/4  ó õ p 0   tan2( y)tan( x) dxdy

Use the properties of the double integrals and the double integrals

R  f( x,y) dA = 5        S  f( x,y)dA = 7        R  g( x,y) dA = 11

to evaluate the double integrals below:

23. R 7f( x,y) dA 24. R [ f(x,y) -g( x,y) ] dA 25. R [ f( x,y) +2g( x,y) ] dA 26. R [ f( x,y) -3f( x,y) ] dA 27. RÈS  f( x,y) dA 28. RÈS 7f( x,y) dA 29. RÈS  g( x,y) dA - S  g( x,y) dA 30. RÈS [ f( x,y) +g( x,y)] dA - S  g( x,y) dA

31. Find the volume of the solid bound between the surfaces z = x²+y² and z = 9.

32. Find the volume of the solid bound between the surfaces z = x²+y² and z = 2x. (hint: integrate over the region whose boundary curve is the intersection of the two surfaces).

33. Show that for all ( x,y) in [ 0,1]×[ 0,1] that

0 £  sin( px) 1+cos2( y) £ sin( px)

and then use this result to estimate

ó õ 1 0  ó õ 1 0   sin( px) 1+cos2(y) dydx

34. Let D denote the unit circle. Explain why

D  ex+ydA £ ó õ 1 -1  ó õ 1 -1  ex+ydydx

and then evaluate this last integral.

35. Suppose that f( x) ³ 0 over [ a,b] and recall that the surface of revolution obtained by revolving the graph of f about the x-axis is given by

r( u,v) = á v,f( v) cos(u) ,f( v) sin( u) ñ

for u in [ 0,2p] and v in [ a,b] . Show that the volume of the resulting solid of revolution is

ó õ b a  ó õ f( x) -f( x)   [f( x) ] 2-y2   dydx

and then compute the innermost integral using the trigonometric substitution

y = f( x) sin( q)

36. Suppose that f( x) > 0 for all x in (a,b) and suppose that f( a) = f( b) = 0. What is the volume of the solid enclosed by the surface

y2+z2 = [ f( x) ] 2

37. Use the Riemann definition of the double integral to prove (3).

38. Use the Riemann definition of the double integral to prove (1).

39. Write to Learn: Suppose that f( x,y) is integrable over two bounded, non-overlapping regions R and S. Let g₁( x,y) = f( x,y) if ( x,y) is in R and g₁( x,y) = 0 if ( x,y) is not in R. Similarly, let g₂( x,y) = f( x,y) if (x,y) is in S and let g₂( x,y) = 0 otherwise. Write a short essay in which you show that

RÈS  f( x,y) dA  =  [ a,b] ×[c,d]  [ g1( x,y) +g2( x,y) ] dA

where [ a,b] ×[ c,d] contains RÈS. Then in that essay use this result to prove (8).

40. Write to Learn: Write a short essay in which you show that

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