College of the Holy Cross, Spring Semester, 2005

Math 132, Makeup Midterm 2 April 1

Name:

Instructions Please write your answers in the spaces provided, and show work on the test itself. Use the back of the preceding page if you need more space for scratch work.

Please do not write in the space below

Problem Points/Poss

1  / 16

2  / 12

3  / 18

4  / 14

5  / 14

6  / 14

7  / 12

Total  /100

1. The graph of the function f(x) is shown below.

0 1 2 3 4 5 6 7 8 123456789 10

(a) [10 points] Use the graph of f to compute the following approximations of ∫_{0}^{8} f(x) dx. LEF T (2) : RIGHT (2) : MID(2) : T RAP (2) : SIMP (2) :

(b) [6 points] For each method, decide whether the approximation of ∫_{0}^{8} f(x) dx is an overestimate, an underestimate, or that this cannot be determined from the given information. In the case of an overestimate or underestimate, briefly explain your reasoning.

Method Type of Estimate Reason

LEF T RIGHT MID T RAP SIMP 2. (a) [6 points] Set up an integral that represents the arclength of the portion of the

graph of y = cos(x) between x = 0 and x = π. Do not compute the integral.

(b) [6 points] Approximate the integral in part (a) using a left hand sum with n = 4 subintervals.

Math 132, Midterm 2 4

3. Rewrite each of the following improper integrals as a limit, or limits. State whether the integral converges or diverges, and compute its value if it converges.

(a) [6 points] ∫_{0}^{3} \frac{1}{\sqrt{9 − x^{2}}} dx

(b) [6 points] ∫_{−1}^{2} \frac{1}{x^{3/2}} dx

(c) [6 points] ∫_{0}^{∞} e^{−3x} dx 4. Let R denote the region bounded by y = sin x, y = cos x, x = 0 and x = π/4.

0  π/4 1  R  x y

(a) [7 points] Find the area of R.

(b) [7 points] Find \overline{x}, the x-coordinate of the center of mass of R. (Assume the region has constant mass density δ = 1.)

5. Let R be the region bounded by the curve y = e^{x} and the lines x = 0 and y = e.

e y R  x

(a) [8 points] Find the volume of the solid obtained by revolving R about the x-axis.

(b) [6 points] Set up an integral that represents the volume of the solid whose base is R and whose cross-sections perpendicular to the x-axis are semicircles with base lying in R. Do not compute the integral.

6. Suppose a metal rod of length 2 meters has a mass density δ(x) = 5 + 2x kg per meter. (a) [7 points] Find the total mass of the rod.

(b) [7 points] Find the center of mass of the rod.

7. A 200 meter tall pyramid has a square base with side length 100 meters. Its mass density (in kg/m^{3}) at height h meters above the base is given by δ(h) = 1 − .0005h.

100 h 200

(a) [6 points] Write down a Riemann sum that approximates the total mass of the pyramid.

(b) [6 points] Using the answer to part (a), write down an integral that equals the mass of the pyramid. Do not compute the integral.