Math 132: Calculus for Physical and Life Sciences 2 Answers for Problem Set 6 Due Friday, March 14, 2008, at the beginning of class.

1. (a) Cross sections are disks with radius x^{2} + 2. Volume of solid is \frac{293π}{15}.

(c) The cross-section by a plane x = c is a washer with inner radius 2 − sin(x) and outer radius 2. The volume is 8π − \frac{π^{2}}{2} .

(e) The solid generated rotating around the x-axis is the union of two solids meeting only along the plane x = \sqrt{2}/2. The cross-sections of the first are the washers with inner radius x and outer radius 2x. The cross-sections of the second are also washers with inner radius x and outer radius 1/x. The total volume is \frac{4π}{3}(\sqrt{2}−1).

(g) The cross-section by a plane y = d is a washer with inner radius \frac{1}{y} and outer radius 2. The volume equals \frac{49π}{4}.

2. Answer: 36\sqrt{3}.

3. Answer: 40000π cubic feet.

4. (a) The graph is a straight line from the point (3, −2) to the point (11, 4). The arclength is 10.

(c) The curve is the arc of a circle with radius 3 and center (0, 0) starting at (0, 3) and proceeding clockwise to (0, −3) (one half of the full circle). The arclength is 3π.

5. Arclength ≐ 3.343018.

6. (a) Answer: \frac{−2}{9π}.

8. Answer: 3.44 units.