16 MULTIPLE INTEGRALS

16 SAMPLE EXAM
Problems marked with an asterisk (*) are particularly challenging and should be given careful consideration.
1. Consider the function f (x, y) = x y on the rectangle [1, 2] × [1, 2].
(a) Approximate the value of the integral ∫ 2∫ 2
1 1
f (x, y) dydx by dividing the region into four squares
and using the function value at the lower left-hand corner of each square as an approximation for the
function value over that square.
(b) Does the approximation give an overestimate or an underestimate of the value of the integral? How
do you know?
2. Given that ∫ π/2
0
dx
1 + sin 2 x = π
2 √ 2 ,
(a) evaluate the double integral
(b) evaluate the triple integral
3. Consider the rugged region below:
∫ π/2 ∫ π/2
0
0
1
( 1 + sin 2 x )( 1 + sin 2 ) dx dy
y
∫ π/2 ∫ π/2 ∫
(
1/ 1+sin 2 y )
0
0
1/ ( 1+sin 2 x ) dzdx dy
(a) Divide the region into smaller regions, all of which are Type I.
(b) Divide the region into smaller regions, all of which are Type II.
4. Rewrite the integral
in rectangular coordinates.
∫ 2π ∫ 1
0 0
932
r 2 dr dθ