Math 152 Final Exam

If u = g(x) is a differentiable function, then
∫f(g(x))g′(x) dx = ∫

=∫f(u) du

Integrals of the form:
∫(f(x))ⁿ*f′(x) dx:
Let u =

Let u = f(x)

Integrals of the form:
∫e^f(x)*f′(x) dx:
Let u =

Let u=f(x)

Integrals of the form:
∫f'(x) / f(x) dx
Let u=

Let u=f(x)

Integrals of the form:
∫cos(f(x))f′(x) dx
∫sin(f(x))f′(x) dx
∫sec²(f(x))f′(x) dx
∫csc²(f(x))f′(x) dx
∫sec(f(x)) tan(f(x))f′(x) dx
∫csc(f(x)) cot(f(x))f′(x) dx

Let u=

Let u=f(x)

Area:
A=∫

A = ∫[Top − Bottom] dx if an x-integral is preferred

A=∫[Right to Left] dy if a y-integral is perfered

Area:
If the curve intersects between the points of intersection, then...

then the integral must be split into parts to calculate the full area on either side of the cruve

Volume:
Disk around the x-axis

V = ∫(f(x))² dx

Volume:
Disk around the y-axis

V = ∫pi* (g(y))² dy

Volume:
Disk around a line other than the x or y axis

The radius must be adjusted accordingly

Volume:
Washer around the x-axis

V = ∫((f(x))² − (g(x))²) dx, where f(x) ≥ g(x) for a≤ x≤b

Volume:
Washer around the y-axis

V = ∫(g(y))² − (h(y))²) dy where g(y) ≥ h(y) for c≤ y≤ d

Volume:
Washer revolving around a line other than the x or y axis

The inner and outer radius must be adjusted accordingly

Volume:
Shell around the x-axis

V = ∫2πyg(y) dy

Volume:
Shell around the y-axis

V=∫2πx(f(x)) dx

Volume:
Cylinderical shell revolving around a line other than the x or y-axis

Adjust the radius of the shell accordingly

Slicing:
⊥ to x-axis and A is the area of the cross section

V=∫Adx

Slicing:
⊥ to y-axis and A is the area of the cross section

V∫Ady

Work:
Force is constant

W=Fd

Work:
Force is not constant

W=∫f(x) dx

Work:
Spring Problem
f(x) is the force needed to maintain it past natural length
W is the work to keep it beyond its natural length

f(x)= kx
W=∫f(x) dx

Work:
Rope pulling Problem
Rope is b units long and weights w, the work to pull the entire rope

W=∫(0 to b) w*x*dx

Work:
Rope pulling Problem
Rope is b units long and weights w, the work to pull part of the rope up

The total work required to pull the rope plus the work required to lift the weight

Work:
Water pumping Problem
pg is the weight density
A is the area of a slice of water
d is the distance traveled to leave the tank

W=pg ∫ dA dy

Average Value:
f(x) from x=a to x=b

f(ave) = 1/(b-a) ∫f(x) dx

Integration by Parts:
∫udv

uv-∫vdu

Integration by Parts:
Order of Importance

Inverse trig, Log, Algebraic functions, Trig functions, Exponential functions

Trig Integrals:
Sin and Cos
When one or both are odd

Factor out one of the odd,
sin odd, u=cos(x)
cos odd, u=sin(x)

Trig Integrals:
Sin and Cos
Both are even

sin²x=½(1-cos(2x))
cos²x=½(1+cos(2x))

Trig Integrals:
Sec and Tan
tan odd

Factor our one sec(x)tan(x),
Use tan²x=sec²x−1
u=sec(x)

Trig Integrals:
Sec and Tan
sec is even

Factor out a sec²(x),
Use sec²x=1+tan²(x)
u=tanx

Trig Sub:
a²-x²

x=a*sinθ

Trig Sub:
a²+x²

x=a*tanθ

Trig Sub:
x²-a²

x=a*secθ

Trig Sub:
Form of √ax²+bx+c

Must complete the square to solve

Partial Fractions:
Usable when...

when f(x) is in the form f(x) = g(x) / h(x) when h(x) is a higher degree

Partial Fraction:
Linear factors

(x+1) /(x-2)(2x+11) = A/(x-2) + B/(2x-11)

Partial Fractions:
Repeating linear factors

(x+1) / (x-2)² = A/(x-2) + B/(x-2)²

Partial Fractions:
Irreducible quadratic factors

(x+1)/(x-2)(x²+1) = A/(x-2) + Bx+C / (x²+1)

Improper Integrals:

take the limits of the integral, if there is a point of discontinuity the integral needs to be separated into it's parts

Comparison Theory for Improper Integrals

If ∫f(x) dx converges, so does ∫g(x) dx
If ∫g(x) dx diverges, so does ∫f(x)dx

Arc Length:
If y=f(x)

L=∫√1+(f'(x))² dx

Arc Length:
If c=g(y)

L=∫√1+(g'(y))² dy

Arc Length:
If x=f(t)

L=∫√(f'(t))² + (g'(t))² dt

Surface Area of Revolution:
around x-axis

SA=2π ∫f(x) √1+(f'(x))² dx

SA=2π ∫y √1+(g'(x))² dx

SA=2π ∫g(t) √(f'(t))² + (g'(t))² dx

Surface Area of Revolution:
around y-axis

SA=2π ∫x √1+(f'(x))² dx

SA=2π ∫g(y) √1+(g'(x))² dx

SA=2π ∫f(t) √(f'(t))² + (g'(t))² dx

Sequences

An infinite, ordered list of numbers

Sequence:
Limit of sequence

lim (as n goes to ∞) = finite number or infinite

Sequence:
Bounded sequence

Must be bounded above and below

Sequence:
Increasing or decreasing

Increasing aⁿ < aⁿ⁺¹
Decreasing aⁿ > aⁿ⁺¹

Sequence:
Recursive sequence

a sequence where a¹ is given, and aⁿ⁺¹ = f(aⁿ). First, find the first few terms of the sequence to get a feel for whether the sequence converges. Then, to find the limit, take the limit of both sides of the recursive definition.

Series

The sum of a sequence

Partial Sums of a Series

s¹ = a¹
s²= a¹ + a²
s³= a¹ + a² + a³
etc. till
sⁿ= a¹ + a² + a³ + ... + aⁿ

Series:
sum

the sum is s,
finite - converges
infinite - diverges

Geometric Series

∑ (1 to ∞) arⁿ⁻¹ converges if |r| < 1 and diverges if |r| ≥ 1.
If |r| < 1, then the sum is ∑ (1 to ∞) arⁿ⁻¹ = a/(1 − r)

Telescoping Series

∑ (1 to ∞)aⁿ⁺i − aⁿ) for some integer i ≥ 1
The series 'collapses'. To find the sum, find a formula for sⁿ and then find lim sⁿ

Test of Divergence

If the lim of the series doesn't equal zero, then the series diverges

Integral Test

If f(x) is a positive, continuous, decreasing function on [k,∞], where k is a non-negative integer, and aⁿ = f(n). Then
∑aⁿ and ∫f(x) dx either both
converge or both diverge

If the improper integral converges, so does the series. If
the improper integral diverges, so does the series

Comparison Test

Suppose ∑aⁿ and ∑bⁿ are series of positive terms

If ∑bⁿ is convergent and aⁿ ≤ bⁿ for all n, then ∑aⁿ is convergent
- If the larger series converges, so does the smaller series

If ∑bⁿ is divergent and aⁿ ≥ bⁿ for all n, then ∑aⁿ is also divergent
-If the smaller series diverges, so does the larger series

Limit Comparison Test

If lim (n→∞) aⁿ/bⁿ = c > 0, then
either both series converge or both series diverge

Remainder Estimate

Rⁿ = S-sⁿ

Altering Series Test

∑(-1)ⁿaⁿ, where aⁿ > 0, conver
satisfies:
• aⁿ⁺¹ ≤ a¹ (sequence {aⁿ} is decreasing)
• lim n→∞ aⁿ = 0