SAMPLE MIDTERM II

MATH 228: Calculus III
SAMPLE MIDTERM II
Spring 2014
NAME :
NOTE : There are 5 problems on this midterm (total of 6 pages). Use of
calculators will NOT be permitted. In order to receive full credit for any
problem, you must show work leading to your answer. You have 50 minutes
to complete this test.
Problem Possible
points
1
20
2
20
3
20
4
20
5
20
Total
100
Score
MATH 228 – MIDTERM II
Page 2
Problem 1. (20pts)
(a) (10pts) Sketch the region in the xy-plane whose area is given by the
iterated integral
Z 6 Z 2y
dx dy
0
y 2 /3
(b) (10pts) Evaluate the iterated integral.
MATH 228 – MIDTERM II
Page 3
Problem 2. (20pts)
(a) (10pts) Change the Cartesian integral into an equivalent polar integral.
Z 2Z x
y dy dx
0
0
(b) (10pts) Evaluate the polar integral.
MATH 228 – MIDTERM II
Page 4
Problem 3. (20pts) Setup and evaluate an intreated integral for 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).
MATH 228 – MIDTERM II
Page 5
Problem 4. (20pts) E is the solid right circular cylinder whose base is the
circle r = 2 sin θ in the xy-plane and whose top lies in the the plane z = 4−y.
Setup an iterated integral for evaluating (BUT DO NOT EVALUATE)
Z Z Z
z dV
E
MATH 228 – MIDTERM II
Page 6
Problem 5. (20pts) Let E be the smaller cap cut from a solid ball of radius
2cm by a plane a distance 1cm from the center of the sphere. Compute the
volume of E.