College of the Holy Cross, Fall Semester, 2004 Math 132, Midterm 1 (All Sections) Wednesday, February 23, 6 PM

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  / 20

2  / 10

3  / 10

4  / 20

5  / 20

6  / 10

7  / 10

Total  /100

Math 132, Midterm 1 2

1. [10 points each] Determine whether each of the following improper integrals converges, and if so, find its value.

(a) ∫_{0}^{∞} e^{−4x} dx

(b) ∫_{0}^{1} \frac{dx}{\sqrt{1 − x}}

Math 132, Midterm 1 3

2. [10 points each] Each part of this question concerns the region R in the plane bounded by the graph y = \frac{1}{6}x^{3} + \frac{1}{2}x^{−1}, the x-axis, and the lines x = 1, x = 4.

(a) Find x̄, the x coordinate of the center of mass of R, and mark it in the figure below.

0  1  2  3  4 048 12 16  x y  x̄ =

(b) Find the volume of the solid generated when R is revolved about the x-axis.

Volume:

Continued next page (c) Find the arc length of the curve along the top of R.

Arc length:

Math 132, Midterm 1 5

3. [5 points each] Each part concerns the integral ∫_{0}^{3}(2 − \frac{1}{x − 4}) dx.

0 1 2 3 012

(a) Compute LEFT(3) and RIGHT(3).

LEFT(3) RIGHT(3)

(b) Compute TRAP(3) and MID(3).

TRAP(3) MID(3)

(c) Compute the exact value of the integral.

Exact value:

(d) Use parts (b) and (c) to compute the error for the trapezoid and midpoint formulas with three subintervals.

Trapezoid error: Midpoint error: