College of the Holy Cross, Spring 2008 Math 132, Midterm Exam 2 (All Sections) Wednesday, March 26, 6 PM

Your Name:

Your Section:

Little (8:00am) Ballantine (9:00am) DeStefano (10:00am) DeStefano (noon)

Instructions: For full credit, you must show all work on the test pages and place your final answer in the box provided for the problem. Use the back of the preceding page if you need more space for scratch work. The numbers next to each part of the questions are their point values.

Please do not write in the space below

Problem Points/Poss
I  / 15
II  / 25
III  / 30
IV  / 15
V  / 15
Total  /100

A. (5) Use midpoint Riemann sums with n = 4 to approximate ∫_{1}^{2} \sqrt{1 + ln(x)} dx.

Midpoint approximation:

B. (5) Use the trapezoidal rule with n = 4 to approximate ∫_{1}^{2} \sqrt{1 + ln(x)} dx.

Trapezoidal rule approximation:

C. (5) The function y = \sqrt{1 + ln(x)} is plotted below.

Check the appropriate boxes:

The midpoint approximation is a overestimate □/underestimate □.

The trapezoidal rule approximation is a overestimate □/underestimate □.

II. For each of the following improper integrals, set up and evaluate the appropriate limits to determine whether the integral converges. If so, find its value; if not, say “does not converge.” (Credit will be given only for the correct limit calculation.)

A. (8) ∫_{1}^{∞} e^{−x} dx

B. (8) ∫_{−1}^{1} \frac{1}{x^{2}} dx C. (9) ∫_{0}^{1} x ln(x) dx.

4 III. The region R is bounded by the graphs y = \sqrt{x} and y = x^{2}.

A. (5) Sketch the region R.

B. (5) Set up and compute an integral to find the area of R.

Area =

C. (10) The region R is rotated about the x-axis to generate a solid. Set up and compute an integral to find its volume.

Volume = D. (10) The region R is rotated about the y-axis to generate a solid. Set up and compute

an integral to find its volume.

Volume =

(go to next page)

IV. (15) Set up and compute the integral for the arclength of the curve x = 5t^{2} +2, y = t+1, 0 ≤ t ≤ 2. (You may consult the table of integrals for this one; if you do, say which table entry you are using.)

V. (15) Find the average value of f(x) = \frac{x^{3}}{\sqrt{25 − x^{2}}} on the interval 0 ≤ x ≤ 2. (Use a trigonometric substitution to compute the integral you need.)