TEXAS A&M UNIVERSITY
DEPARTMENT OF MATHEMATICS
MATH 251-507/509
Exam 3 version A, 12 Nov 2009
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.
Sketch the domain of integration and evaluate
ZZ
(x2 + y 2)3/2 dA,
D
where D is the region in the first quadrant bounded by x = 0, y =
√
3x and x2 + y 2 = 4.
2.
Find the area of the surface of the part of the sphere x2 + y 2 + z 2 = 4 that lies above the
p
cone z = x2 + y 2.
Hint: Use the “area of the surface” formula
ZZ q
1 + [fx ]2 + [fy ]2 dA
A(S) =
D
3.
Find the mass of the pyramid with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 2) and the
density proportional to the distance from the XZ plane.
4.
Evaluate the integral
ZZZ
z 2 dV,
E
2
where E is the solid outside the cylinder x +y 2 = 1 and inside the ellipsoid x2 +y 2 +4z 2 = 5.
5.
Bonus question +2 points (give explanations!): Find and sketch the region E for
which the triple integral
ZZZ
E
(1 − x2 − 2y 2 − 3z 2 ) dV
is maximal.
Points:
/20