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Calculus II Cheat Sheet by

Master cheat sheet for Calculus II. Cheat sheets for Integral Trigonometry and Conic Sections are sourced. Integral Trigonometry Cheat Sheet: https:­//c­hea­tog­rap­hy.c­om­/cr­oss­ant­/ch­eat­-sh­eet­s/i­nte­gra­l-t­rig­ono­metry/ Conic Sections Cheat Sheet: https:­//c­hea­tog­rap­hy.c­om­/cr­oss­ant­/ch­eat­-sh­eet­s/c­oni­c-s­ect­ions/


Infinite Series
Σaₙ=Σₙ₌ₖ aₙ
Converges if lim n->∞ Sₙ=L
Diverges if lim n->∞=∞ or DNE
Sₙ is the partial sum of the series: Sₙ=a₁+­a₂+­a₃+­a₄+...+­aₙ­₋₁+aₙ
Harmonic Series
Never converges
Always diverges
The altern­ating version of this series (Σ(-1)n+1/n) converges, and Σ1/n is a P-Series with p=1
Geometric Series
Σₙ₌₀ arnₙ₌₁ arn-1
Converges if |r|<1
Diverges if |r|≥1
If the series converges, its sum is S=a/(1-r)
Converges if p>1
Diverges if p≤1
Altern­ating Series
Σ(-1)nbₙ, Σ(-1)n+1bₙ, or Σ(-1)n-1bₙ
Converges if lim n->∞=0 and bₙ is a decreasing sequence (bₙ₊₁≤bₙ for all n)
Cannot show diverg­ence, inconc­lusive
Telesc­oping Series
Converges if lim n->∞=L
Diverges if lim n->∞ Sₙ=∞ or DNE
Sₙ is the partial sum of the series: Sₙ=Σⁿᵢ­₌₁(­bᵢ-­bᵢ₊ₙ) where n is finite
Altern­ating Series Estimation Theorem: If Sₙ=Σⁿᵢ­₌₁(­-1)ⁿbₙ or Σⁿᵢ₌₁(­-1)­ⁿ⁻¹bₙ is the sum of an altern­ating series that converges, then |Rₙ|=|­S-S­ₙ|≤bₙ₊₁
Trigon­ometric functions like cos(nπ) or sin(nπ/2) act as sign altern­ators, like (-1)n
The Altern­ating Series Test does not show diverg­ence, however, implem­enting the test requires a Test For Diverg­ence, which does show divergence

Series Tests

Test for Divergence
Cannot show conver­gence, inconc­lusive
Diverges if lim n->∞≠0
Integral Test
Σaₙ=f(n), which is a positive, contin­uous, decreasing function on the interval [k,∞), usually with easily­-in­teg­rable functions
Converges if ∫ₖf(n)dn converges
Diverges if ∫ₖf(n)dn diverges
(Direct) Comparison Test
aₙ and bₙ are positi­ve-­termed (aₙ≥0 and bₙ≥0 for all n) and aₙ≤bₙ for all n
Σaₙ converges if Σbₙ converges
Σbₙ diverges if Σaₙ diverges
Inconc­lusive if bₙ diverges or aₙ converges
Limit Comparison Test
aₙ and bₙ are positi­ve-­termed, and lim n->∞ aₙ/bₙ=c, or lim n->∞ bₙ/aₙ=d, where c and d are finite constants greater than 0
Σaₙ converges ⟺ Σbₙ converges
Σaₙ diverges ⟺ Σbₙ diverges
Inconc­lusive if either c or d=0 or ∞
Ratio Test
Σaₙ, usually with n! terms or aₙn
Absolutely converges if lim n->∞ |aₙ₊₁/­aₙ|­<1
Diverges if lim n->∞ |aₙ₊₁/­aₙ|­>1
Inconc­lusive if lim n->∞ |aₙ₊₁/­aₙ|=1
Root Test
Σaₙ, usually with aₙn
Absolutely converges if lim n->∞ |aₙ|1/n<1
Diverges if lim n->∞ |aₙ|1/n>1
Inconc­lusive if lim n->∞ |aₙ|1/n=1
Absolu­te/­Con­dit­ional Conver­gence
Absolutely converges if Σ|aₙ| converges
Condit­ionally converges if Σ|aₙ| diverges, but Σaₙ converges
Diverges if Σaₙ diverges
For the series listed, assume each series to be an infinite series starting at n=k: Σₙ₌ₖ=Σ
If Test for Divergence passes (lim n->­∞=0), use another test
The symbol [ ⟺ ] represents the relati­onship "if and only if" (often abbrev­iated to "­iff­"), meaning both sides of the statement must be true at the same time, or false at the same time
If a test is inconc­lusive, use another test

Special Series

Power Series centered at a
Taylor Series centered at a
Maclaurin Series (Taylor Series centered at 0)
R=1, I=(-1,1)
R=∞, I=(-∞,∞)
R=1, I=(-1,1]
R=1, I=[-1,1]
R=∞, I=(-∞,∞)
R=∞, I=(-∞,∞)
Taylor's Inequality
|Rₙ(x)­|≤M­|x-a|n+1/(n+1)!, given M≥|f(n+1)(x)| for all |x-a|≤d
For the series listed, assume each series to be an infinite series starting at n=0: Σₙ₌₀=Σ
Note that the formula for a Degree 1 Taylor Polyno­mial, T₁(x), has the same formula as the Linear Approx­imation formula learned in Calculus I
f(n) means "the nth derivative of the function f"
n! = n(n-1)­(n-­2)(­n-3­)...*3*2*1
0!=1, 1!=1

Areas of Functions

Between two functions
∫ₐᵇ ((top functi­on)­-(b­ottom functi­on))dA
Enclosed by a polar function
½∫ₐᵇ f(θ)2
Between two polar functions
½∫ₐᵇ ((outer polar function)2-(inner polar function)2)dθ
Area enclosed by a polar function is with respect to the pole, which is the origin

Average value of a function: f
=1/(b-­a)∫ₐᵇ f(x)dx

Integr­ation by Parts

Indefinite Integral
Definite Integral
∫ₐᵇ udv=uv|ₐᵇ -∫ₐᵇ vdu
Integr­ation by Parts is used to integrate integrals that have components multiplied together in their simplest form, often referred to as a "­product rule for integr­als­"

The order in which to choose the "­u" variable often goes by the acronym LIPET: Logari­thms, Inverse trigon­ometry, Polyno­mials, Expone­ntial functions, then Trigon­ometry

The constant of integr­ation does not need to be inserted until the integral has been fully simplified

Trigon­ometric Integrals

Conic Sections


Volumes of Solids of Revolution

π∫ₐᵇ (radius)2dV
π∫ₐᵇ (outer radius)2 - (inner radius)2dV
Cylind­rical Shell
2π∫ₐᵇ (radiu­s)(­hei­ght)dV
For Cylind­rical Shells: radius=x or y, and height­=f(x) or g(y)

Integr­ation by Partial Fractions

A/(x-a) + B/(x-b)
A/(x-a) + B/(x-a)2
A/(x-a) + B/(x-b) + C/(x-c)
A/(x-a) + B/(x-a)2 + C/(x-b)
A/(x-a) + Bx+C/(x2+bx+c)
∫1/(a2+x2) dx
Integr­ation by Partial Fractions is used to simplify integrals of polynomial rational expres­sions into simpler fractions with a factored, irredu­cible denomi­nator

The degree (highest power) of the numera­tor's polynomial must be less than the degree of the denomi­nator's polyno­mial, otherwise, polynomial long division must be used before converting the expression into partial fractions

Parametric Curves and Polar Functions

Parametric Curve C as a function of Parameter t
(x,y)=­(f(­t),­g(t)) for t on [a,b]
Slope at a given point
Second derivative
Polar Curve C as a function of Parameter θ
Slope at a given point
Cartes­ian­/Re­cta­ngular to Polar coordi­nates
x=rcos(θ), y=rsin(θ)
Polar to Cartes­ian­/Re­cta­ngular coordi­nates
r2=x2+y2 or r=(x2+y2), tanθ=y/x or θ=arct­an(y/x)
(dx/dt)≠0, (dx/dθ)≠0

Arc Lengths

∫ₐᵇ (1+(f'(x))2)dx
Parametric Function
∫ₐᵇ ((x'(t))2+(y'(t))2)dt
Polar Function
∫ₐᵇ (r(θ)2+(r'(θ))2)dθ
For standard functions: f'(x)=­dy/dx
For parametric functions: x'(t)=­dx/dt and y'(t)=­dy/dt
For polar functions: r'(θ)=­dr/dθ

Integral Approx­ima­tions and Error Bounds

Midpoint Rule
Trapez­oidal Rule
Simpson's Rule
Midpoint Rule Error Bound
|Eₘ|≤k­(b-a)3/24n2, k=f''(­x)ₘₐₓ on [a,b]
Trapez­oidal Rule Error Bound
|Eₜ|≤k­(b-a)3/12n2, k=f''(­x)ₘₐₓ on [a,b]
Simpson's Rule Error Bound
|Eₛ|≤k­(b-a)5/180n4, k=f(4)(x)ₘₐₓ on [a,b]
Integral Approx­ima­tions are typically used to evaluate an integral that is very difficult or impossible to integrate


x̄=(xᵢ­₋₁+­xᵢ)/2, the averag­e/m­edian of two points xᵢ₋₁ and xᵢ

Simpson's Rule can only be used if the given n is even

In order of most accurate to least accurate approx­ima­tion: Simpson's Rule, Midpoint Rule, Trapez­oidal Rule, Left/Right endpoint approx­imation

Surface Areas

Function revolved about an axis
2π∫ₐᵇ (radiu­s)(Arc Length compon­ent)ds
Function revolved about y-axis
2π∫ₐᵇ x(1­+(f­'(x))2)dx
Function revolved about x-axis
2π∫ₐᵇ y(1+(g'(y))2)dy
Parametric function of t revolved about y-axis
2π∫ₐᵇ f(x)((x'(t))2+(y'(t))2)dt
Parametric function of t revolved about x-axis
2π∫ₐᵇ g(y)((x'(t))2­+(y'(t))2)dt
f'(x)=­dy/dx, g'(y)=­dx/dy, x'(t)=­dx/dt, and y'(t)=­dy/dt

Improper Integrals

∫ₐ f(x)dx
lim t->∞ ∫ₐᵗ f(x)dx
∫_-∞ᵇ f(x)dx
lim t->-∞ ∫ₜᵇ f(x)dx
∫_-∞ f(x)dx
lim t->-∞ ∫ₜᶜ f(x)dx + lim t->∞ ∫꜀ᵗ f(x)dx
Conver­gence of ∫f(x)dx
lim t->±∞=L
Divergence of ∫f(x)dx
lim t->­±∞=±∞ or DNE
Improper Integrals are integrals with bounds at infinity (Type 1) or discon­tinuous bounds (Type 2)

[ _-∞ ] represents negative infinity as a lower bound


Hey everyone! I noticed some errors in the Series Tests, apparently from a wrongful copy-and-paste that slipped by. If you downloaded this sheet before this comment was posted, please re-download it now that the errors are corrected. Thanks and happy studying! - CROSSANT

CROSSANT CROSSANT, 06:03 18 Dec 23

Hey that's me

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