Math 172
Exam II
Last Name(print):
March 22, 2001
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You must show all appropriate work to receive full credit.
Work all problems on the paper provided. Turn in your exam with your work.
Do not work on the back on any page. SCHOLASTIC DISHONESTY WILL NOT BE TOLERATED.
Integrals
that might be helpful.
Z
Z
sec xdx = ln | sec x + tan x| + C
csc xdx = ln | csc x − cot x| + C
1
1
sec3 xdx = sec x tan x + ln | sec x + tan x| + C
2
2
Z
−1
1
3
csc xdx =
csc x cot x + ln | csc x − cot x| + C
2
2
Z
1. (8 points) An artificial lake has been constructed to have the shape in the figure. Measurements
across the the lake are taken at 25 foot increments and are indicated on the diagram. Use
Simpson’s rule to estimate the surface area of the lake.
72ft
38ft
45 ft
59ft
2. (16 points) Evaluate the integral or show that it is divergent.
Z∞
(a)
1
Z3
(b)
−2
−1
dx =
(2x + 1)(3x + 1)
1
dx =
x2
3. (6 points) A system has four points with masses m1 = 3, m2 = 7, m3 = 2 and m4 = 4. The
first three points are located at P1 (1, 5), P2 (−3, 2), and P3 (−1, −4). What are the coordinates
of point P4 (x, y) so that the centroid is located at (1, 2)?
4. (8 points) What is the exact length of the curve x = et − t, y = 4et/2 for 0 ≤ t ≤ 2?
5. (8 points) Compute the surface area of the surface obtained by rotating the curve x = 1 + 2y 2
for 1 ≤ y ≤ 2 about the x-axis.
More on the back
6. (9 points) Solve the differential equation.
dy
= xex+2y
dx
7. A tank contains 20 lb of sugar mixed with 50 gal of water. Sugar water containing 3 lb of sugar
per gal is added to the tank at the rate of 7 gal per min. The tank is kept thoroughly mixed
and drains at the same rate.
(a) (5 points) Write out the differential equation and the initial condition for y(t), the number
of lbs of sugar in the tank at time t.
(b) (6 points) Solve the initial value problem.
(c) (4 points) How much sugar is in the tank after half an hour?
8. (14 points) Does the sequence converge or diverge? If it converges, find what it converges to.
Show work to justify your answer.
√
1 − 3n
√
(a) an =
1 + 5n
(b) a1 = 3 an+1 = 8 − an for n ≥ 1
n
o∞
9. (6 points) Is the sequence n2 − 7n + 10
work to justify your answer.
n=6
increasing, decreasing or not monotonic? Show
10. (10 points) An isosceles triangular plate is suspended vertically in water as shown. The height
of the triangle is 1.5 meters, its base is 2 meters, and it is submerged 1 meter into the water.
Find the force on one face of the plate. Shaded area is the water.
1.5 m
1m
2m