- (12 points) By reversing the order of integration, find the exact value of
Z 8
0
Z 2
√3 y
e
x
4
dxdy
Steps:
(a) Based upon the limits in the integrated integrals, express the region D algebraically.
(b) Sketch the integration region D.
(c) Reverse the order of the given iterated integrals.
(d) Evaluate the new iterated integrals.
1 - (12 points) Consider the iterated integral
Z 1
0
Z x
0
2xy dydx +
Z 2
1
Z 2−x
0
2xy dydx.
(a) Based upon the limits in the integrated integrals, express the regions D1 and D2
algebraically.
(b) Sketch both D1 ad D2 in the same xy-plane.
(c) Change the order of integration.
(d) Evaluate the integral.
2 - (12 points) Set up, but do not evaluate, iterated integrals that gives the volume
between the plane z = 4 and the salad bowl z = x
2 + y
2
.
(a) Find the intersection of the plane z = 4 and the salad bowl z = x
2 + y
2
. The
projection of the solid E between the plane z = 4 and the salad bowl z = x
2 + y
2
is the integration region D.
(b) Sketch the region D and expression it algebraically.
(c) Set up the iterated integrals that computes the volume E
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