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# Calculus II Cheat Sheet by CROSSANT

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: Σₙ₌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."

### Volumes

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.

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