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MA2255-003/Calculus III/Spring 2012
Test 4 (50 minutes)
Name (UPPER CASE):
(Sections 15.1, 15.2, 15.3, 15.4, 15.4, 15.5)
Please read and follow the instruction for each problem carefully. Show your work
in detail. No credit will be given if detailed computational, algebraic and/or graphic
arguments are not provided.
1. (4+4+4=12 points) Consider the iterated integral
π
S0 S0
x
sin y
dy dx.
y
(a) Sketch the region of integration.
(b) Reverse the order of integration.
(c) Evaluate the integral.
2. (12 points) Find the volume of the solid beneath the paraboloid z x2 y 2 and above
the triangle enclosed by the lines y x, x 0, and x y 2 in the xy-plane. [Hint: Express
the volume as a double iterated integral.]
1
3. (4+8=12 points)
(a) Sketch the region bounded by the curve y
ex and the lines x
0, y
0, and x
ln 2.
(b) Express the area of the bounded region in part (a) as an iterated double integral and
evaluate the integral.
º
4. (12 points) Find the centroid of the semicircular region bounded by the x-axis and the
1 x2 .
curve y
2
5. (6+6=12 points) Consider the integral
0
0
S1Sº1x2
1
»2 2
x
y2
dy dx.
(a) Change the Cartesian integral into an equivalent polar integral.
(b) Evaluate the polar integral obtained in part (a).
6. (4+6+4= 14 points) Let D be the region bounded by the paraboloids z
and z x2 y 2 .
8 x2 y 2
(a) Write the iterated triple integral for the volume of D with the order dz dy dx.
(b) Evaluate the integral in part (a).
(c) Write the iterated triple integral for the volume of D with the order dx dz dy.
3
7. (14 points) Find the volume of the solid region in the first octant bounded by the coordinate planes and the plane y z 2 and the cylinder x 4 y 2 .
4
8. (12 points) Find the center of mass of the solid of constant density bounded below by
the paraboloid z x2 y 2 and above by the plane z 4. [Hint: The integrals involved are
triple iterated integrals.]
Extra-credit Problem (5 points)
Find the average height of the hemisphere z
»
a2 x2 y 2 over the circular disk
x2 y 2 B a2
in the xy-plane.
5