Vv156 Honors Calculus II

Assignment 5

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 = cos x, y = 2 − cos x, 0 ≤ x ≤ 2π. x = 2y

2

, x = 4 + y

2

(ii) .

(2 pts)

Exercise 5.2 [Ste10, p. 427] Evaluate the integral and interpret it as the area of a region. Sketch the region.

Z π/2

0

(i) |sin x − cos 2x| dx

Z 1

−1

|3

x − 2

x

(ii) | dx

(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) (1 pt) y = x

3

, y = 8, x = 0; about y = 0.

(b) (1 pt) x = 4y

2 − y

3

, x = 0; about y = 0.

(c) (1 pt) y = x

4

, y = 0, x = 1; about x = 2.

(3 pts)

Exercise 5.5 [Ste10, p. 453]

(a) (2 pts) If f is continuous and Z 3

1

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

[1, 3].

(b) (2 pts) Find the numbers b such that the average value of f(x) = 2 + 6x − 3x

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

Z

f(x) dx = xf(x) −

Z

xf′

(x) dx

1

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

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(b) (2 pts) If f and g are inverse functions and f

′

is continuous, show that

Z b

a

f(x) dx = bf(b) − af(a) −

Z 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 Z e

1

ln x dx.

(8 pts)

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

Z π

−π

(i) sin mx cos nx dx = 0.

Z π

−π

sin mx sin nx dx =

(

0, if m ̸= n

π, if m = n

(ii)

Z π

−π

cos mx cos nx dx =

(

0, if m ̸= n

π, if m = n

(iii)

(3 pts)

Exercise 5.8 [Ste10, p. 478] A finite fourier sine series is given by the sum f(x) = X

N

n=1

an sin nx. Show that the mth

coefficient am is given by

am =

1

π

Z π

−π

f(x) sin mx dx

(2 pts)

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

Z ∞

0

dx

√

x(1 + x)

(i) Z ∞

0

ln x

1 + x

2

(ii) dx

(4 pts)

Exercise 5.10 [Ste10, p. 543] Find the exact length of the curve.

(i) y = ln(sec x), 1 ≤ x ≤ 2. y = 3 + 1

2

(ii) cosh 2x, 0 ≤ x ≤ 1.

(2 pts)

Exercise 5.11 [Ste10, p. 544] Find the arc length function for the curve y = arcsin x +

p

1 − x

2 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.

y = x

3

(i) , 0 ≤ x ≤ 2. 9x = y

2

(ii) + 18, 2 ≤ x ≤ 6.

(2 pts)

Exercise 5.13 [Ste10, p. 573] Let f(x) = 30x

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) (1 pt) Find P(X ≤

1

3

).

(2 pts)

Exercise 5.14 [Ste10, p. 573] Let f(x) = c/(1 + x

2

).

(a) (1 pt) For what value of c is f a probability density function?

(b) (1 pt) 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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