MATH 2110 – EXAM 4 REVIEW PROBLEMS
This is not a comprehensive set of exercises. Be sure to review the homework
and notes as well when preparing for the exam.
1. Evaluate
2
3 9
2 2 3/ 2
0 0
( ) dy dx
x
x y
−
+
∫ ∫
by converting to polar coordinates
2. Evaluate
2 2
R
x y dA
+
∫∫
where R = {(x, y) |
2 2
9, 0
x y x
+ ≤ ≥
}
3. Find the mass and center of mass of the lamina that is bounded by the triangular region
(0, 0), (4, 3), (4, 0) with density
( , )
x y c
ρ
=
.
4. Evaluate
4 1
0 0 0
sin dz dy dx
x
x y
π
−
∫∫ ∫
5. Use a triple integral to find the volume of the tetrahedron in the first octant bounded by
the coordinate planes and the plane passing through (1, 0, 0), (0, 2, 0), and (0, 0, 3).
6. Convert
( 1, 2 , 4)
− to cylindrical coordinates
7. Convert
( 3,1,2 3)
to spherical coordinates
8. Identify each surface:
a.)
2
ρ
=
b.) z =
2
r
c.) r = 4
9. Use a triple integral to find the volume of the solid bounded by the paraboloids
2 2
z x y
= +
and
2 2
24 5 5
z x y
= − − in the first octant.
10. Use spherical coordinates to evaluate
2 2 2 2
( )
x y z
E
xe dV
+ +
∫∫∫
where E is the solid that lies
between the spheres
2 2 2
9
x y z
+ + =
and
2 2 2
16
x y z
+ + =
in the first octant.
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