Mathematics 103, Section 208, Final Review
April 8, 2013
Pooya Ronagh (pooya@math.ubc.ca)
Problem 1. Write a left and a right Riemann sum for
b
∫
x2 dx
b
and
∫
a
x3 dx
a
and evaluate them.
1
Problem 2. Calculate using Riemann sums: ∫0 ex dx
Problem 3. Evaluate
2b/n + 22b/n + ⋯ + 2(n−1)b/n + 2nb/n
n→∞
n
lim
3
4 1/8
∫ x (1 − 12x ) dx
∫
5
∫
ln x
dx
x
t1/2 ln(2t)dt
1
dx
∫
x ln x
cos x
dx
∫
sin3 x
n
3 23
lim ∑ (1 + i. )
n→∞
n n
i=1
x
dx
∫
x + 10
5
∫
t1/2 ln(t)dt
1
√
8 + 9xdx
∫
9
∫ sec x tan xdx
6
2
∫ tan x sec xdx
2b
3 − 5 + 7 − 9 + 11 − ⋯ − 81
2
∫
(3x + 5)n dx
0
∫
x2 + 2x + 3
dx
(x + 1)(x + 2)(x + 3)
π
sin x
dx
∫
0 (2 + cos(x))3
xdx
+ b2
b
1+ε
1
1
√
lim ∫
dt
ε→0 ε 1
1 + t4
√
e
ln x
dx
∫
x
1
19
dx
∫
2
18 x − 34x + 289
∫
x2
Problem 4. Prove V = Sh/3 for a pyramid with (a) a square base, (b) a circular base, (c)
a shape as base! Here S denotes the area of the base. (Hint: you need to use the fact that
if you rescale an object by a factor of s, the area also rescales by a factor of s2 .)
√
x
Problem 5. Show that y(x) = ∫0 1 − t2 dt satisfies the differential equation
y ′ y ′′ = −x;
y ′ (0) = 1
y(0) = 0,
Problem 6. Let a > 0. Show that over positive x-axis,
x
∫
0
√
dt
√
= ln(x + x2 + a2 ) − ln a
t2 + a2
Find c such that
x
∫
c
√
dt
√
= ln(x + x2 + a2 ).
t2 + a2
x
Problem 7. If f (x) is continuous, find f (π/2) if ∫0 f (t)dt = 2x(sin x + 1). Solve the same
x/2
problem if ∫0 f (t)dt = 2x(sin(x) + 1).
Problem 8. Find the area between
y = 4 − x2 ,
x = y 2 − 4y,
y = 3x
x = 2y − y 2 .
Problem 9. Find area under y = 1 − x2 in two ways.
Problem 10. Find volume of solid of revolution of the region bounded by
y = 2ax − x2 , y = 0 about both coordinate axis
√
y = ax, y = 2, x = a about both coordinate axis
y = (x − 1)2 , y = (x + 1)2 , y = 0 about both coordinate axis
y = ex , y = 2, x = 0, about y = −1.
Problem 11. Find average distance from a point on the perimeter of a square of side
length a to the center. Find the average of the square of the distance.
Problem 12. Find arclength
y = 5x + 2, 0 ≤ x ≤ 1
y = x3/2 , 0 ≤ x ≤ 1
y = (ex + e−x )2, 0 ≤ x ≤ b
x = t2 , y = t3 , 0 ≤ t ≤ 2
x2 /a2 + y 2 /b2 = 1
Problem 13. Solve the following differential equations and initial value problems
dy
= (2x + 5)4
dx
dy
1
=
dx y + 1
dy
= xy 2
dx
cos x sin y dy = sin x dx,
Problem 14. Find y(5) if y satisfies
dy
dx
= x2 y −1 ,
y(0) = 0
y(0) = 10.
Problem 15. If X1 and X2 have uniform distribution between zero and 2, Let X = X1 +X2 .
Find the distribution, mean, median and variance of X.
Problem 16. A point is chosen randomly (and uniformly) on a unit circle in the plane.
Find the distribution of the projection of this point on the x-axis. Find its expected value
(i.e. mean).
Problem 17. A router receives two types of request from client computers with exponential distributions X1 and X2 of respective parameter λ1 and λ2 . Find the distribution in
time for the stand-by type of the router. (Hint: For intervals I and J on the real line we
may assume P (X1 ∈ I and X2 ∈ J) = P (X1 ∈ I).P (X2 ∈ J).)
Problem 18. Determine value of the improper integral if it converges or prove that it
diverges:
∞
∫
1
∞
sin(x)dx
∫
0
∞
∫
1
π/4
dx
x ln x
tan xdx
∫
−π/2
∞
∫
dt
t2 − 2t + 1
0
dt
√
t t2 − 1
Problem 19. Determine the divergence and convergence of the following series
3n−1
3n+1
n=2 2
∞
∑
en
n=1 n!
∞
∑
∞
∞
∞
∑
∑
4 − 3 sin(2n)
n2 − 1
n=2
−n2
∑ ne
n=3
√
∞
nn!
∑
2
n=10 [(2n)!]
n=1
(n4
1
− 1)1/3
Problem 20. Find Taylor series expansion of the following functions, and the interval of
their convergence.
sin(x)/x, sin(x) + cos(x)
Problem 21. Find a 3-rd degree approximation of ln(3/2).
Problem 22. Write the Taylor polynomial of degree 5 for ln(x) about a = 1
degree n for cos(x) + 3 about a = 0
degree 4 for y = x2 + x − 1 about origin (a = 0)
degree 4 for y = x2 + x − 1 about x = 10.