Mathematics 134 – Intensive Calculus for Science 2 Exam 2 – March 24, 2006

Directions: Do all work in the blue exam booklet. You may use an approved calculator, and the table of integrals provided, as directed. There are 100 possible points.

I. Compute each of the integrals below, using some combination of integration by substitution, parts, and partial fractions. You must show all work for full credit. Applying an entry from the integral table is OK if necessary, but that will result in a deduction of 3 points per part.

A) (15)

∫ x cos(4x) dx

B) (15)

∫ \frac{x + 3}{x^{2} − 6x + 8} dx

II. Compute each of the following integrals using an entry or entries from the table of integrals. Say which entry or entries you are using.

A) (10)

∫ sin^{4}(x) dx

B) (10)

∫ (x^{2} + 12)e^{3x} dx

III. (20) Let R be the region bounded by y = x^{2} + 2, y = x, x = 0 and x = 1. Find the volume of the solid obtained if R is rotated about the line y = 5.

IV. A long thin straight wire extends from x = 0 to x = 1 cm. The density of the wire at x is δ(x) = 2 + cos(πx) grams per cm. A) (10) Set up, but do not evaluate the formula computing the center of mass of the wire. B) (10) If the midpoint Riemann sum with n = 10 is used to approximate the integral

for the total mass, will the result be an overestimate or an underestimate? Explain.

V. (10) The graph y = f(x) below shows either a probability density (pdf) or a cumulative distribution (cdf). Say which it is, and find the value of c given in the graph.