A:
Math 2E
Mid-term Exam I
Name:
Student ID:
Discussion session:
8 am
2 pm
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circle one
This exam consists of 4 questions and 100 total points. The point value of each problem is indicated. Read
directions for each problem carefully. Please show all work needed to arrive at your solutions. Clearly
indicate your nal answers.
Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10 points)
Find the volume of the solid that lies between the paraboliod z = x2 +y 2 and the sphere x2 +y 2 +z 2 = 6.
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Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10 points)
Find the volume of the solid that lies between the cone z = 3x2 + 3y 2 and the sphere x2 + y 2 + z 2 = 1.
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Problem 3 RR
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10 points)
Evaluate R (y + 3x)2 dA, where R is bounded by y = 1 − 3x; y = 3 − 3x; y = x − 1; and y = x − 3.
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Problem 4 R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10 points)
Evaluate C 3xds, where C is the line segment C1 from (0, 0) to (1, 0), followed by C2 : the quarter circle
to (0, 1).
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Problem 5 p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10 points)
Let f = x2 + y 2
a) Find the gradient vecotr of F~ = ∇f
b) Using the gradient vector in a), compute
points
Q1
Q2
Q3
Q4
R
C
F~ · d~r where C is a line segment from (1, 1) to (2, 2).
Q5
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