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

General Directions: You must show all work for credit on these problems.

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

(a) y = x^{2} + 2, x = 1, x = 2, y = 0; about the x-axis

(b) y = \frac{1}{2}x + 3, y = x^{2}; about the x-axis

(c) y = sin x, x = 0, x = π, y = 0; about the line y = 2

(d) y = x^{3}, y = \sqrt{x}, x = 0, x = 1; about the x-axis

(e) y = \frac{1}{x}, y = x, y = 2x for x > 0; about the x-axis

(f) x = 4 − y^{2}, x = y^{2}; about the y-axis

(g) x = \frac{1}{y}, y = 4, x = 2; about the y-axis

2. The base of a certain solid is the circle x^{2} +y^{2} = 9 and each cross section perpendicular to the x-axis is an equilateral triangle with one side across the base. Find the volume of the solid.

3. A nose cone for a space reentry vehicle is designed so that a cross section, taken x feet from the the tip and perpendicular to the axis of symmetry, is a circle of radius \frac{1}{4}x^{2} ft. Find the volume of the nose cone given that its length is 20 ft.

4. Graph the curve and find its exact length.

(a) x = 4t + 3, y = 3t − 2, 0 ≤ t ≤ 2

(b) x = \frac{1}{3}t^{3}, y = \frac{1}{2}t^{2}, 0 ≤ t ≤ 1

(c) x = 3 sin(2t) and y = 3 cos(2t), 0 ≤ t ≤ \frac{π}{2}

(d) y = 3x^{2} + 2 and 2 ≤ x ≤ 4

5. Use Simpson’s Rule with n = 10 to estimate the arc length of the curve y = ln x for 1 ≤ x ≤ 4.

6. Find the average value of the function on the given interval.

(a) y = x cos(3x), [0, π]

(b) y = x\sqrt{1 + 3x^{2}}, [1, 3]

7. Find the number b such that the average value of f(x) = x^{3} − 1 on the interval [0, b] is equal to 6.

8. The amount of a certain drug present in a patient’s body for the first 4 days after the

drug has been administered is

C(t) = 5e^{−0.2t}

units. Determine the average amount of the drug present in the patient’s body for the first 4 days after the drug has been administered.