Math 166

Fall 2001

Exam 2

1. (24 pt) Evaluate the following integrals (or show that they do not converge).

a) b) c)

2. (10 pt) Let Show that converges.

3. (15 pt) Use the arc length integral to show that the circumference of a circle of radius is given by

4. (15 pt) Find the surface area obtained when the parabola is revolved about the axis. What happens to your answer as What should happen?

5. (24 pt) Consider the function For this problem, you can use the fact that

a) Find the volume obtained when the region bounded by and the axis is revolved about the axis.

b) Find the volume obtained when the region bounded by and the axis is revolved about the axis.

c) Use your results from a) and b) to locate the centroid of the region bounded by and the axis.

6. (10 pt) Suppose you want to use Simpson’s rule to estimate Find the minimal so that using gives an error bound less than or equal to .05. The fourth derivative of is pictured below.

7. (12 pt) Water is contained in a glass aquarium in the shape of a cube. Each side of the aquarium is 2 feet long. Find the force due to hydrostatic pressure exerted on each side of the aquarium. (Remember that force is pressure times area and that in our system of units, pressure is depth times 62.5).