10th - d should show what the variables are in the...

INTEGRATION PROBLEMS, MATH  , WEEK 
DAVID PIERCE
 . Find
R
f when R = [0 , 2] × [1 , 2] and f ( x, y ) = 3 xy 2 - 2 x 2 y .
 . Find the average value of y 2 / ( x 2 + 1) when 0 x 1 and 0 y 1 .
 . Express as an iterated integral the volume enclosed by:
• the two sheets of the hyperboloid given by 4 x
2
- 4 y
2
- z
2
= 4 ;
• the XY - and XZ -planes;
• the planes given by z = 2 and y = 3 .
 . Find
R
f when R = { ( x, y ): 0 y π & 0 x 1 + cos y } and f ( x, y ) = x 2 sin y .
 . Write
R
f as an iterated integral in two ways when R is the quadrilateral with vertices
(0 , 0) , (4 , 0) , (3 , 2) , and (0 , 3) .
 . Reverse the order of integration in
e
1
ln y
0
f ( x, y ) d x d y .
 . Compute
2
0
4
x
2
12 x 3 exp( y 3 ) d y d x .
 . Using symmetry, find the volume enclosed by:
• the hyperbolic paraboloid x = y 2 - z 2 ;
• the plane x = 1 ;
• the cylinder y 2 + z 2 = 1 .
Remark. Where Stewart writes
R
f ( x, y ) d A , I write
R
f here, on the basis that:
(a) considered as integrals, both areas and volumes are the same thing: they are limits
of sums of sizes of rectangular figures;
(b) the number of signs d (if used at all) should match the number of signs ;
(c) what comes after the sign