# TEXAS A&amp;M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 251-509 ```TEXAS A&amp;M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 251-509
Exam 3 version A, 11 Nov 2008
On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work.
Name (print):
In all questions, no analytical work — no points.
1.
For the integral
Z
1
1/2
Z
2
yexy dydx,
1/x
(a) sketch the domain of integration, (b) exchange the order of integration and (c) evaluate
the integral.
2.
Find the volume of the body inside both the cylinder x2 + y 2 = 9 and the ellipsoid
4x2 + 4y 2 + z 2 = 100.
3.
Evaluate the integral
ZZZ
(x2 + y 2)dV,
H
where H is the hemispherical region that lies above the xy-plane and below the sphere
x2 + y 2 + z 2 = 1.
4.
Evaluate the integral
ZZ
(y 2 − x2 )dA,
D
where D is the region between the lines y = x, y = x − 2, y = −x and y = 2 − x. (Hint:
while it can be done by other means too, changing the variables makes the integral very
easy).
5.
Bonus question +10%: Find
z = 0, y + z = 1 and x + z = 1.
RRR
E
zdV , where E is bounded by the planes x = 0, y = 0,
Points:
/20
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