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Vv156 Honors Calculus II  Assignment 5

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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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