Description

This assignment has a total of (40 points).
Note: Unless specified otherwise, you must show the details of your work via logical reasoning for each exercise. Simply writing a final result (whether correct or not) will receive 0 point.
Exercise 5.1 [Ste10, p. 427] Sketch the region enclosed by the given curves and find its (unsigned) area.
(i) y = cosx, y = 2 − cosx, 0 ≤ x ≤ 2π. (ii) x = 2y2, x = 4 + y2.
(2 pts)
Exercise 5.2 [Ste10, p. 427] Evaluate the integral and interpret it as the area of a region. Sketch the region.

(2 pts)
Exercise 5.3 [Ste10, p. 440] Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.
(2 pts)
Exercise 5.4 [Ste10, p. 445] Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis.
(a) (1pt) y = x3, y = 8, x = 0; about y = 0.
(b) (1pt) x = 4y2 − y3, x = 0; about y = 0.
(c) (1pt) y = x4, y = 0, x = 1; about x = 2.
(3 pts)
Exercise 5.5 [Ste10, p. 453]
(a) (2pts) If f is continuous and , show that f takes on the value 4 at least once on the interval
[1,3].
(b) (2pts) Find the numbers b such that the average value of f(x) = 2+6x−3×2 on the interval [0,b] is equal to
3.
(4 pts)
Exercise 5.6 [Ste10, p. 470]
(a) (2pts) Use integration by parts to show that
Z Z
f(x)dx = xf(x) − xf′(x)dx
(b) (2pts) If f and g are inverse functions and f′ is continuous, show that

(c) (2pts) In the case where f and g are positive functions and b > a > 0, draw a diagram to give a geometric interpretation of part (b).
(d) (2pts) Use part (b) to evaluate .
(8 pts)
Exercise 5.7 [Ste10, p. 478] Prove the formula, where m and n are positive integers.

(3 pts)
N
Exercise 5.8 [Ste10, p. 478] A finite fourier sine series is given by the sum f(x) = X an sinnx. Show that the mth
n=1 coefficient am is given by
(2 pts)
Exercise 5.9 [Ste10, p. 528] Evaluate the integral
Z ∞ dx (i) √
0 x(1 + x)
(4 pts)
Exercise 5.10 [Ste10, p. 543] Find the exact length of the curve.
(i) y = ln(secx), 1 ≤ x ≤ 2. .
(2 pts)

p 2 with starting point Exercise 5.11 [Ste10, p. 544] Find the arc length function for the curve y = arcsinx + 1 − x
(0,1).
(2 pts)
Exercise 5.12 [Ste10, p. 550] Find the exact area of the surface obtained by rotating the curve about the x-axis.
(i) y = x3, 0 ≤ x ≤ 2. (ii) 9x = y2 + 18, 2 ≤ x ≤ 6.
(2 pts)
Exercise 5.13 [Ste10, p. 573] Let f(x) = 30×2(1 − x)2 for 0 ≤ x ≤ 1 and f(x) = 0 otherwise.
(a) (1pt) Verify that f is a probability density function.
(b) (1pt) Find .
(2 pts)
Exercise 5.14 [Ste10, p. 573] Let f(x) = c/(1 + x2).
(a) (1pt) For what value of c is f a probability density function?
(b) (1pt) For that value of c, find P(−1 < X < 1).
(2 pts)
References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on pages 1, 2).
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