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Math 181, Calculus II Hour Exam Two 9:00 am April 11, 1997

1. The number of bird nests in a coastal wetland is approximately f(x) = x(6 − x) nests per square mile for 0 ≤ x ≤ 6 where x is the distance in miles from the coast. a. Set up an integral that represents the total number of nests in a rectangular area 20 miles along the coast and 6 miles inland from the coast to dryer ground.

b. Evaluate this integral.

2. Find the volume of the solid obtained by rotating the region bounded by the lines

y = 3 − x, y = 1, x = 0, and x = 1

about the x-axis.

3. Some gasoline is stored in a cylindrical container 1.5 feet high and 3 feet in circumference. Recall that the pressure of the gasoline at a depth of y feet from the top of the container is 42y lb/ft^{2}. Find the total force on the side on the container.

4. The probability distribution p(x) is given by

p(x) = \frac{2}{9}x for 0 ≤ x ≤ 3.

a. What is the probability that x lies between 1 and 2?

b. What is the median value of x? c. What is the mean value of x?

5. Find the Taylor polynomial of degree 4 approximating

f(x) = x − ln(1 + x) for x near 0.

Use your answer to find

lim_{x→0} \frac{x − ln(1 + x)}{x^{2}}.