Calculus II Cheat Sheet

Series

Infinite Series	Σaₙ=Σ^∞ₙ₌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	Σ1/n	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	Σ^∞ₙ₌₀ arⁿ=Σ^∞ₙ₌₁ ar^n-1	Converges if \|r\|<1	Diverges if \|r\|≥1	If the series converges, its sum is S=a/(1-r)
P-Series	Σ1/n^p	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	Σ(bₙ-bₙ₊₁)	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	Σaₙ	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 ∫ₖ^∞f(n)dn converges	Diverges if ∫ₖ^∞f(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	Σaₙ	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: Σ^∞ₙ₌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	ΣCₙ(x-a)ⁿ	C₀+C₁(x-a)+C₂(x-a)²+C₃(x-a)³+C₄(x-a)⁴+...
Taylor Series centered at a	Σf⁽ⁿ⁾(a)(x-a)ⁿ/n!	f(a)+f'(a)(x-a)+f''(a)(x-a)²/2!+f'''(a)(x-a)³/3!+f⁽⁴⁾(a)(x-a)⁴/4!+...	R>\|x-a\|
Maclaurin Series (Taylor Series centered at 0)	Σf⁽ⁿ⁾(0)(x-0)ⁿ/n!=Σf⁽ⁿ⁾(0)xⁿ/n!	f(0)+f'(0)x+f''(0)x²/2!+f'''(0)x³/3!+f⁽⁴⁾(0)x⁴/4!+...
1/(1-x)	Σxⁿ	1+x+x²+x³+x⁴+...	R=1, I=(-1,1)
e^x	Σxⁿ/n!	1+x+x²/2!+x³/3!+x⁴/4!+...	R=∞, I=(-∞,∞)
ln(1+x)	Σ(-1)ⁿ⁺¹xⁿ/n	x-x²/2+x³/3-x⁴/4+x⁵/5+...	R=1, I=(-1,1]
arctan(x)	Σ(-1)ⁿx²ⁿ⁺¹/(2n+1)	x-x³/3+x⁵/5-x⁷/7+x⁹/9+...	R=1, I=[-1,1]
sin(x)	Σ(-1)ⁿx²ⁿ⁺¹/(2n+1)!	x-x³/3!+x⁵/5!-x⁷/7!+x⁹/9!+...	R=∞, I=(-∞,∞)
cos(x)	Σ(-1)ⁿx²ⁿ/(2n)!	1-x²/2!+x⁴/4!-x⁶/6!+x⁸/8!+...	R=∞, I=(-∞,∞)
(1+x)^k	Σ(^kₙ)xⁿ=Σ((k(k-1)(k-2)(k-3)...(k-n+1))/n!)xⁿ	1+kx+k(k-1)x²/2!+k(k-1)(k-2)x³/3!+k(k-1)(k-2)(k-3)x⁴/4!+...	R=1
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: Σ^∞ₙ₌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(n-1)(n-2)!=n(n-1)(n-2)(n-3)!=...
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)²dθ
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

avg

=1/(b-a)∫ₐᵇ f(x)dx

Integration by Parts

Indefinite Integral	∫udv=uv-∫vdu
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

https://cheatography.com/crossant/cheat-sheets/integral-trigonometry/

Conic Sections

https://cheatography.com/crossant/cheat-sheets/conic-sections/

Volumes of Solids of Revolution

Disk	π∫ₐᵇ (radius)²dV
Washer	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

(px+q)/((x-a)(x-b))	A/(x-a) + B/(x-b)
(px+q)/(x-a)²	A/(x-a) + B/(x-a)²
(px²+qx+r)/((x-a)(x-b)(x-c))	A/(x-a) + B/(x-b) + C/(x-c)
(px²+qx+r)/((x-a)²(x-b))	A/(x-a) + B/(x-a)² + C/(x-b)
(px²+qx+r)/((x-a)(x²+bx+c))	A/(x-a) + Bx+C/(x²+bx+c)
∫1/(a²+x²) dx	(1/a)arctan(x/a)+C

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	dy/dx=(dy/dt)/(dx/dt)
Second derivative	d²y/dx²=(dy/dt)/(dx/dt)²
Polar Curve C as a function of Parameter θ	(r,θ)=(r,θ±2πn)=(-r,θ±πn)
Slope at a given point	dy/dx=(dy/dθ)/(dx/dθ)
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

Function	∫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	Δx(f(x̄₁)+f(x̄₂)+f(x̄₃)+...+f(x̄ₙ₋₁)+f(x̄ₙ))
Trapezoidal Rule	(Δx/2)(f(x₁)+2f(x₂)+2f(x₃)+...+2f(xₙ₋₁)+f(xₙ))
Simpson's Rule	(Δx/3)(f(x₁)+4f(x₂)+2f(x₃)+4f(x₄)+2f(x₅))+...+2f(xₙ₋₂)+4f(xₙ₋₁)+f(xₙ))
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

Δx=(b-a)/n

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

Created By

CROSSANT

Metadata

Languages: English

Published: 28th May, 2023
Last Updated: 26th March, 2024

Comments

CROSSANT, 02:35 3 Jun 23

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, 06:03 18 Dec 23

Hey that's me

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Series

Series Tests

Special Series

Areas of Functions

Integration by Parts

Trigonometric Integrals

Conic Sections

Volumes of Solids of Revolution

Integration by Partial Fractions

Parametric Curves and Polar Functions

Arc Lengths

Integral Approximations and Error Bounds

Surface Areas

Improper Integrals

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Calculus II Cheat Sheet by CROSSANT

Series

Series Tests

Special Series

Areas of Functions

Integr­ation by Parts

Trigon­ometric Integrals

Conic Sections

Volumes of Solids of Revolution

Integr­ation by Partial Fractions

Parametric Curves and Polar Functions

Arc Lengths

Integral Approx­ima­tions and Error Bounds

Surface Areas

Improper Integrals

Created By

Metadata

Comments

Add a Comment

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Latest Cheat Sheet

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About Cheatography

Behind the Scenes

Recent Cheat Sheet Activity

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Integration by Parts

Trigonometric Integrals

Integration by Partial Fractions

Integral Approximations and Error Bounds