Final Exam Review
• Area Between Curves
See §7.1 examples 1, 2, 4, 5 and exercises 1-33 (odd)
The area of the region bounded by the curves y = f (x), y = g(x), and the lines x = a
and x = b, where f and g are continuous and f (x) ≥ g(x) is
b
Z
b
Z
[f (x) − g(x)]dx.
[Top − Bottom]dx =
A=
a
a
The area of the region bounded by the curves x = f (y), x = g(y), and the lines y = c
and y = d, where f and g are continuous and f (y) ≥ g(y) is
Z
d
d
Z
[Right − Left]dy =
A=
c
[f (y) − g(y)]dy.
c
• Volume (Method of Disks or Washers)
See §7.2 examples 1-6 and exercises 1-11, 25-37 (odd)
The Disk Method
b
Z
Z
2
b
[f (x)]2 dx.
π[r(x)] dx = π
V =
a
a
The Washer Method
Z b
Z b
2
2
V =
π [R(x)] − [r(x)] dx = π
[f (x)]2 − [g(x)]2 dx.
a
a
• Volume (Method of Cylindrical Shells)
See §7.3 examples 1-4 and exercises 1-25 (odd)
The Shell Method
Z
V =
b
Z
2πr(x)h(x)dx = 2π
a
b
xf (x)dx.
a
• Work
See §7.4 examples 1-4 and exercises 1-19 (odd), 20
The work done by a variable force, F (x), in moving an object from x = a to x = b is
Z
W =
b
F (x)dx.
a
• Average Value of a Function
See §7.5 example 1 and exercises 1-5 (odd)
The average value of f (x) on the interval [a, b] is
1
Avg. =
b−a
Z
b
f (x)dx.
a
• Integration by Parts
See §8.1 examples 1-5 and exercises 1-33 (odd)
Integration by parts formula
Z
Z
udv = uv −
vdu.
• Trigonometric Integrals
See §8.2 examples 1-9 and exercises 1-31 (odd)
Integrals of the form
Z
sinn x cosm xdx.
1. If n is odd, factor out one sin x, use the identity sin2 x = 1 − cos2 x, and use the
substitution u = cos x.
2. If m is odd, factor out one cos x, use the identity cos2 x = 1 − sin2 x, use the
substitution u = sin x.
3. If both n and m are even, use the identities
1
sin2 x = (1 − cos 2x)
2
and
1
cos2 x = (1 + cos 2x).
2
Integrals of the form
Z
tann x secm xdx.
1. If m is odd, factor out sec x tan x, use the identity tan2 x = sec2 x − 1, and use
the substitution u = sec x.
2. If n is even, factor out sec2 x, use the identity sec2 x = 1 + tan2 x, and use the
substitution u tan x.
3. If m is even and n is Rodd, use the identity tan2 x = sec2 x − 1 and use integration
by parts to evaluate secn xdx, where n is odd.
• Trigonometric Substitution
See §8.3 examples 1, 3-5, 7 and exercises 1-27 (odd)
Integration by trigonometric substitution
√
1. √
If the integrand involves a2 − x2 , let x = a sin θ, dx = a cos θdθ, so that
a2 − x2 = a cos θ.
√
2. √
If the integrand involves x2 − a2 , let x = a sec θ, dx = a sec θ tan θdθ, so that
x2 − a2 = a tan θ.
√
3. √
If the integrand involves x2 + a2 , let x = a tan θ, dx = a sec2 θdθ, so that
x2 + a2 = a sec θ.
• Partial Fractions
See §8.4 examples 1-7 and exercises 1-51 (odd)
To integrate a rational function
R(x) =
P (x)
,
Q(x)
where P (x) and Q(x) are polynomials, and the degree of P (x) is less than the degree
of Q(x), decompose R(x) into the sum of partial fractions.
1. If Q(x) is the product of distinct linear factors
A
B
C
x2 + 2x − 1
=
+
+
.
(x + 1)(x − 1)(x + 2)
x+1 x−1 x+2
2. If Q(x) is the product of repeated linear factors
x2 + 2x − 1
A
B
C
D
=
+
+
+
.
(x + 1)3 (x + 2)
x + 1 (x + 1)2 (x + 1)3 x + 2
3. If Q(x) contains distinct irreducible quadratic factors
x2 + 2x − 1
Ax + B
B
C
D
= 2
+
+
+
.
2
2
2
(x + 2)(x + 1) (x + 2)
x +2
x + 1 (x + 1)
x+2
4. If Q(x) contains repeated irreducible quadratic factors
x2 + 2x − 1
Ax + B
Cx + D
E
= 2
+ 2
+
.
2
2
2
(x + 2) (x + 1)
x +2
(x + 2)
x+1
If the degree of P (x) is greater than or equal to the degree of Q(x), then we must
perform long division.
• Improper Integrals
See §8.9 examples 1-10 and exercises 3-41 (odd), 49, 51
Summary of improper integrals
1. If f (x) is continuous on [a, ∞), then
Z
Z ∞
f (x)dx = lim
N →∞
a
N
f (x)dx.
a
2. If f (x) is continuous on (−∞, b], then
Z b
Z b
f (x)dx = lim
f (x)dx.
N →−∞
−∞
N
3. If f (x) is continuous on (−∞, ∞), then
Z
Z a
Z ∞
f (x)dx +
f (x)dx =
−∞
−∞
∞
f (x)dx.
a
4. If f (x) is continuous on [a, b) and discontinuous at b, then
Z b
Z N
f (x)dx = lim−
f (x)dx.
N →b
a
a
5. If f (x) is continuous on (a, b] and discontinuous at a, then
Z b
Z N
f (x)dx = lim+
f (x)dx.
N →b
a
a
6. If f (x) is discontinuous at c, where a < c < b, then
Z b
Z c
Z b
f (x)dx.
f (x)dx +
f (x)dx =
a
a
c
• Arc Length
See §9.3 examples 1-3 and exercises 1-9 (odd)
Arc length formulas
1. If we are given a curve with equation y = f (x), a ≤ x ≤ b, then
Z bp
L=
1 + [f 0 (x)]2 dx.
a
2. If we are given a curve with equation x = g(y), c ≤ y ≤ d, then
Z dp
L=
1 + [g 0 (y)]2 dy.
c
3. If we are given a parametric curve defined by x = x(t), y = y(t), α ≤ t ≤ β, then
s
Z β 2 2
dx
dy
L=
+
dt.
dt
dt
α
• Surface Area
See §9.4 examples 1-3 and exercises 1-15, 23-27 (odd)
Surface area formulas
1. If we revolve the curve y = f (x), a ≤ x ≤ b about the x-axis, then
Z b
p
SA = 2π
f (x) 1 = [f (x)]2 dx.
a
2. If we revolve the curve x = g(y), c ≤ x ≤ d about the x-axis, then
Z d p
SA = 2π
y 1 + [g 0 (y)]2 dy.
c
3. If we revolve the parametric curve defined by x = x(t), y = y(t), α ≤ t ≤ β about
the x-axis, then
s 2
Z β
2
dy
dx
y(t)
+
dt.
SA = 2π
dt
dt
α
Similar formulas are used for rotation about the y-axis.
• Sequences
See §10.1 examples 1-11 and exercises 2-25, 37-43 (odd)
A sequence is an ordered list of numbers. A sequence {an } converges with limit L if
lim an = L < ∞.
n→∞
A sequence is bounded if there exists a real number M such that |an | ≤ M for n ≥ 1.
A sequence is decreasing if an+1 < an for n ≥ 1 and increasing if an+1 > an for n ≥ 1.
A sequence that is either increasing or decreasing is called a monotone sequence. Any
bounded, monotone sequence converges.
• Series
See §10.2 examples 1-3, 5-8 and exercises 9-27 (odd)
An infinite series is a sum of the terms of an infinite sequence:
∞
X
n=1
an = a1 + a2 + a3 + . . .
The Divergence Test
If lim an 6= 0, then the series
n→∞
∞
X
an diverges.
n=1
Geometric Series Test
A geometric series is a series of the form
∞
X
arn . The series converges if and only if
n=0
|r| < 1 and its sum is
a
.
1−r
• The Integral Test and Comparison Tests
See §10.3 examples 1-8 and exercises 7-29 (odd)
The Integral Test
Suppose f is a positive, continuous, decreasing function on [1, ∞) and let an = f (n).
Z ∞
∞
X
f (x)dx conThen the series
an converges if and only if the improper integral
1
n=1
verges. If the series converges, then
Z
Z ∞
f (x)dx ≤ Rn ≤
∞
f (x)dx.
n
n+1
The Direct Comparison Test
∞
∞
X
X
Suppose that
an and
bn are series with positive terms.
n=1
n=1
1. If an ≤ bn for all n and
2. If an ≥ bn for all n and
∞
X
n=1
∞
X
bn converges, then
bn diverges, then
n=1
∞
X
an converges.
n=1
∞
X
an diverges.
n=1
The Limit Comparison Test
∞
∞
X
X
Suppose that
an and
bn are series with positive terms and
n=1
n=1
an
= L ∈ (0, ∞).
n→∞ bn
lim
Then either both series converge or both diverge.
• Other Convergence Tests
See §10.4 examples 1-9 and exercises 3-7, 11-13, 17-27 (odd)
Alternating Series Test
If the alternating series
∞
X
(−1)n an satisfies the conditions
n=1
1. lim an = 0
n→∞
2. {an } is a decreasing sequence
then the series converges. If an alternating series converges, then
|Rn | ≤ an+1 .
A series
∞
X
is absolutely convergent if the series
P∞
n=1
|an | converges.
n=1
Ratio Test
Suppose that the series
∞
X
an satisfies
n=1
an+1 = L.
lim
n→∞ an 1. If L < 1, the series converges absolutely.
2. If L > 1, the series diverges.
3. If L = 1, the test fails.
• Power Series
See §10.5 examples 1-5 and exercises 3, 4, 5-17 (odd)
A power series centered at a is a series of the form
∞
X
cn (x − a)n .
n=0
The set of all values of x for which a power series converges is called the interval of
convergence. The radius of convergence is half the length of the interval of convergence.
• Representing Functions as Power Series
See §10.6 examples 1-8 and exercises 3-25 (odd)
If f (x) is defined as a power series
f (x) =
∞
X
n=0
cn (x − a)n ,
with radius of convergence R, then
0
f (x) =
∞
X
n−1
cn n(x − a)
=
∞
X
n=0
n−1
cn n(x − a)
=
n=1
∞
X
cn (n + 1)(x − a)n ,
n=0
Z
f (x)dx = C +
∞
X
cn
n=0
xn+1
.
n+1
The radius of convergence for these series is R.
• Taylor and Maclaurin Series
See §10.7 examples 1-9 and exercises 1-11, 15-21, 33-39, 45-49 (odd)
The Taylor series for f (x) centered at a is
f (x) =
∞
X
f (n) (a)
n=0
n!
(x − a)n .
The Maclaurin series for f (x) is the Taylor series centered at 0.
• Applications of Taylor Polynomials
See §10.9 examples 1-2 and exercises 3-7, 11-19 (odd)
If the Taylor series of f (x) centered at a is
∞
X
f (n) (a)
n!
n=0
(x − a)n ,
then the nth-degree Taylor polynomial of f at a is
Tn (x) =
n
X
f (k) (a)
k=0
n!
(x − a)k .
Taylor polynomials are a polynomial approximation of f (x) for values of x near a.
Taylor’s Inequality provides an upper bound for the error in this approximation:
|Rn (x)| ≤
M
|x − a|n+1 ,
(n + 1)!
where |f (n+1) (x)| ≤ M .
• Three-Dimensional Coordinate System
See §11.1 examples 1-5 and exercises 5-39 (odd)
An equation of the sphere with center (x0 , y0 , z0 ) and radius r is
(x − x0 )2 + (y − y0 )2 + (z − z0 )2 = r2 .
• Vectors and the Dot Product in Three Dimensions
See §11.2 examples 1-6 and exercises 1-31 (odd), 33, 34
The dot product of ~v = hv1 , v2 , v3 i and w
~ = hw1 , w2 , w3 i is the scalar
~v · w
~ = v1 w1 + v2 w2 + v3 w3 .
If θ is the angle between ~v and w,
~ then
cos θ =
~v · w
~
.
||~v ||||w||
~
The vectors ~v and w
~ are orthogonal if and only if ~v · w
~ = 0.
• The Cross Product
See §11.3 examples 3-5 and exercises 7-15 (odd)
The cross product of ~v = hv1 , v2 , v3 i and
~v × w
~ = w
~ = hw1 , w2 , w3 i is the vector
~i ~j ~k v1 v2 v3 .
w1 w2 w3 The cross product is orthogonal to both ~v and w
~ and ||~v × w||
~ is the area of the
parallelogram defined by ~v and w.
~
Note: The best way to study for the exam is to solve the problems from exams in previous
semesters, old homework problems, and as many problems from this review as possible. The
exam will take place May 6th from 8:00-10:00 AM in Blocker 166. Students should
bring a No. 2 pencil, scantron, and valid photo ID. The use of calculators is not allowed.