VV156 Homework 5 Solved

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 = cos x, y = 2 − cos x, 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.

􏰨 π/2 􏰨 1
(i) |sinx−cos2x|dx (ii) |3x −2x|dx

0 −1 (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.1

(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;abouty=0. (b) (1pt)x=4y2−y3,x=0;abouty=0.

(c) (1pt)y=x4,y=0,x=1;aboutx=2. (3 pts)

Exercise 5.5 [Ste10, p. 453]

1. (a)  (2 pts) If f is continuous and

   [1, 3].

2. (b)  (2 pts) 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) (2 pts) Use integration by parts to show that 􏰨􏰨

1 https://en.wikipedia.org/wiki/Steinmetz_solid

􏰨3 1

f (x) dx = 8, show that f takes on the value 4 at least once on the interval

f(x)dx=xf(x)− xf′(x)dx

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(b) (2 pts) If f and g are inverse functions and f ′ is continuous, show that

􏰨 b a

􏰨e 1

(i) (iii)

sinmxcosnxdx = 0.

(ii) sinmxsinnxdx = −π

−π (3 pts)

−π 􏰨π

􏰧0, ifm̸=n π, ifm=n

cos mx cos nx dx =

f (x) dx = bf (b) − af (a) −

􏰨 f(b) f(a)

g(y) dy
(c) (2 pts) 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) (2 pts) Use part (b) to evaluate

(8 pts)

ln x dx.

Exercise 5.7 [Ste10, p. 478] Prove the formula, where m and n are positive integers. 􏰨π 􏰨π

􏰧0, ifm̸=n π, ifm=n

N
Exercise 5.8 [Ste10, p. 478] A finite fourier sine series is given by the sum f (x) = 􏰄 an sin nx. Show that the mth

coefficient am is given by

am = π Exercise 5.9 [Ste10, p. 528] Evaluate the integral

􏰨∞ dx
(i) √x(1+x)

(2 pts)

f(x)sinmxdx −π

􏰨∞lnx
(ii) 1+x2 dx

1􏰨π

n=1

00

(4 pts)
Exercise 5.10 [Ste10, p. 543] Find the exact length of the curve.

(i) y = ln(sec x), 1 ≤ x ≤ 2. (ii) y = 3 + 21 cosh 2x, 0 ≤ x ≤ 1. (2 pts)

Exercise 5.11 [Ste10, p. 544] Find the arc length function for the curve y = arcsin x + 􏰓1 − x2 with starting point (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) (1 pt) Verify that f is a probability density function.

(b) (1pt)Find P(X ≤ 31). (2 pts)

Exercise 5.14 [Ste10, p. 573] Let f(x) = c/(1 + x2).
(a) (1 pt) For what value of c is f a probability density function?

(b) (1pt)Forthatvalueofc,findP(−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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