Show Menu
Cheatography
  1. Home >
  2. Education >

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/

Series

Infinite Series
Σ∞ₙ₌₁ aₙ or Σaₙ
Never converges
Always diverges
Harmonic Series
Σ1/n
Never converges
Always diverges
Geometric Series
Σarn-1
Converges if |r|<1, sum is a/(1-r)
Diverges if |r|≥1
P-Series
Σ(1/n)p=Σ1/np
Converges if p>1
Diverges if p≤1
Altern­ating Series
Σ(-1)naₙ or Σ(-1)n-1aₙ
Converges if lim n->∞=0 AND |aₙ₊₁|­<|aₙ|
Diverges otherwise
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
Note that a Harmonic Series is equal to a P-Series with p = 1, which diverges according to the P-Series test (diverges if p≤1)

Series Tests

Test for Divergence
Σaₙ
Cannot show conver­gence, inconc­lusive
Diverges if lim n->∞≠0
Integral Test
f(n) is positive, contin­uous, and decreasing for all n on the interval [1,∞)
Converges if ∫₁∞f(n)dn converges
Diverges if ∫₁∞f(n)dn diverges
Comparison Test
aₙ < bₙ for all n
Converges if Σbₙ converges
Diverges if Σaₙ diverges
Limit Comparison Test
aₙ and bₙ are comparable
Converges if lim n->∞ aₙ/bₙ=­L>0 and either aₙ or bₙ converge
Diverges if lim n->∞ aₙ/bₙ=­L>0 and either aₙ or bₙ diverge
Ratio Test
Σaₙ
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
Converges if lim n->∞ of the nth root of |aₙ|<1
Diverges if lim n->∞ of the nth root of |aₙ|>1
Inconc­lusive if lim n->∞ of the nth root of |aₙ|=1
Nth root of |aₙ| = |aₙ|1/n
If Test for Divergence passes (lim n->­∞=0), use another test
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+x^4+...
R=1, I=(-1,1)
ex
Σxn/n!
1+x+x2/2!+x3/3!+x4/4!+...
R=∞, I=(-∞,∞)
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=(-∞,∞)
Binomial Series (1+x)k
Σ(nₖ)xn=1+kx+­k(k-1)x2/2!+k(­k-1­)(k-2)x3/3!+...
  
R=1
Taylor's Inequality
|Rₙ|≤M­|x-a|n+1/(n+1)!, given M= f(n+1)(x)ₘₐₓ for all |x-a|≤d
Coeffi­cients (nₖ)
[ k(k-1)­(k-­2)(­k-3­)...(k­-n+1) ]/n!
For the series listed, assume each sum to be an infinite sequence: Σ∞ₙ₌₀ = Σ
Note that the formula for a Degree 1 Taylor Polynomial has the same exact formula as the one used for Linear Approx­imation 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 [f(x)-­g(x)] dx
Polar Function
½∫ₐᵇ f(θ)2dθ
Between Two Polar Functions
½∫ₐᵇ [f(θ)2-g(θ)2] dθ
Area of polar functions is with respect to the origin as the pole.
Average value of a function = 1/(b-a)∫ₐᵇ f(x)dx

Integr­ation by Parts

Indefinite Integral
∫udv= uv-∫vdu
Definite Integral
∫ₐᵇ udv= uv|ₐᵇ -∫ₐᵇ vdu
The order of choosing your "­u" variable goes by LIPET: Logari­thms, Inverse trigon­ometry, Polyno­mials, Expone­ntial functions, then Trigon­ometry.

Integr­ation by Parts is used to integrate integrals that have components multiplied together in their simplest form. It is often referred to as a "­product rule for integr­als."

Trigon­ometric Integrals

Conic Sections

  

Volumes

Disk
π∫ₐᵇ (radius)2 dr
Washer
π∫ₐᵇ (outer radius)2 - (inner radius)2 dr
Cylind­rical Shell
2π∫ₐᵇ (radiu­s)(­height) dr
With respect to the x-axis: use dx
With respect to the y-axis: use dy
For Cylind­rical Shells: radius = x or y, and height = f(x) or f(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. It can only be used when the degree (highest power) of the numerator is less than the degree of the denomi­nator. If the degree of the numerator polynomial is equal to or higher than the degree of the denomi­nator, you must use polynomial long division before converting to partial fractions to integrate.

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, tanθ=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

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) ]
Δx
(b-a)/n
x̄
(xᵢ₋₁+­xᵢ)/2
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]
Simpson's Rule can only be used if the given n is even
In order of most accurate to least accurate: Simpson's Rule, Midpoint Rule, Trapez­oidal Rule
Integral Approx­ima­tions are typically used to evaluate an integral that is very difficult or impossible to integrate.
  

Surface Areas

Function revolved about an axis
2π∫ₐᵇ (radiu­s)(Arc Length component) dx
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 Revolved About Y-Axis
2π∫ₐᵇ f(x)*√[ (x'(t))2+(y'(t))2 ] dt
Parametric Function 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
∫_-∞ᵇf­(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
The symbols "­_∞" and "­_-∞­" represent infinity and negative infinity as a lower bound of an integral.
                               
 

Metadata

  • Languages:
  • Published: 28th May, 2023
  • Last Updated: 12th September, 2023

Comments

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

Add a Comment

Your Comment

Please enter your name.

    Please enter your email address

      Please enter your Comment.

          Related Cheat Sheets

          Weights and Measures Cheat Sheet
          Master Measures & Weights with this Cheat Sheet Cheat Sheet

          More Cheat Sheets by CROSSANT

          Conic Sections Cheat Sheet
          Integral Trigonometry Cheat Sheet