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MA 126 Closed Book/No calculators NAME_________________________________ Show all work for full credit! Spring, 2010
PART 1. Part 1 consists of 8 questions. Do your work and clearly write your answer in the space
provided. (6 points each)
- Set up, but do not evaluate, a definite integral for the area of the region bounded by
y = 9 − x^2 and y = 6 − 2 x. Answer: ________________
- Evaluate the improper integral, or show it diverges. (^2) 0
dx x
∞
Answer: ________________
- Determine in in-lbs how much work is required to stretch a spring 3 inches past its natural length if it takes a force of 10 lbs to hold the spring stretched 2 inches past its natural length.
Answer:____________________
- Evaluate the improper integral 2 3 0
x e xdx
∞ −
∫ , or show that it diverges.
- Which of the following integrals are improper?
A.
2
1
dx
∫ x −
B.
1
0
dx
∫ x −
Answer:_____________________
C. 2
sin 1
x dx x
∞
−∞ +
∫ D.
1
ln( 1)
x
∫^ x^ − dx
- Which pictured area, A or B, is given by
1 2 0
∫ (1^ − y^ ) dy?
Answer:______________________
- Find the arc length of the curve r t ( ) =< 2sin ,3 , 2 cos t t t >, 0 ≤ t ≤ 10
, 0 ≤ t ≤ π.
Answer:______________________
- Find the volume of the solid generated by rotating the region bounded by the curves y = x , y =0,
and x = 1 about the y-axis. Answer: ____________________
Problem 3
Find the length of the curve (^2 ) ( 2 )( 1) y = 3 x + from x = 0 to x = 2. You must find the actual length
here.
Problem 4 A water tank is in the shape of a right circular cone with its vertex down. It is 10 meters in diameter at
the top and 7 meters high. It is filled to a depth of 4 meters with water whose density is 1000^ kg 3 m
Set up a definite integral in terms of one variable and representing the work done in emptying the tank by pumping the water to the top of the tank. Do not evaluate the integral. Sketch the tank and label the part that represents the variable in your integral.
Problem 5
Use the method of slicing (disk method) to set up a definite integral in terms of one variable that represent the volume of the solid generated by revolving the first quadrant region A about each of the given lines below. Do not evaluate the integrals. The region A is bounded by the curves
y = x^2^ , y = 4, and the y-axis. It is revolved about
a. the line y = 4.
b. the line x =2.
