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Calculus II Cheat Sheet by

Master cheat sheet for Calculus II.

Series

Series Type
General Summation
Conver­gence
Divergence
Notes
Infinite Series
Σan=Σ∞n=k an, where k is some whole number (k = {0, 1, 2, ...}) for which an is well-d­efined
Converges if lim n->∞ Sn=L, where L is finite
Diverges if lim n->∞ Sn=∞ 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 arn-1, where a is the first term of the series, and r is the common ratio between terms
Converges if |r|<1
Diverges if |r|≥1
If the series converges, its sum is S=a/(1-r)
P-Series
Σ1/np, where p is a positive number
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->∞ bn =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->∞ Sn=L, where L is finite
Diverges if lim n->∞ Sn=∞ or DNE
Sn is the partial sum of the series: Sn=Σni=1(bi-bi+1) where n is finite
Altern­ating Series Estimation Theorem: If Sn=Σni=1(-1)ibi or Σni=1(-1)i-1bi is the sum of an altern­ating series that converges, then |Rn|=|S-Sn|≤bn+1
Trigon­ometric functions like cos(nπ) or sin(nπ­+π/2) act as sign altern­ators, like (-1)n
The Altern­ating Series Test (AST) checks the limit, but since the AST only concludes conver­gence, we explicitly apply the Test For Divergence (albeit, redund­antly) if the limit fails (even though it checks the same limit)

Series Tests

Test Type
Typical series to use test
Conver­gence
Divergence
Notes
Test for Divergence
Σan
Cannot show conver­gence, inconc­lusive
Diverges if lim n->∞ an≠0 or DNE
Integral Test
an≡f(n), which is a positive, contin­uous, decreasing function on the interval [k,∞), usually with clearl­y-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 in proving conver­gence, or an converges in proving divergence
Limit Comparison Test
an and bn are positi­ve-­termed, and lim n->∞ an/bn=c, where 0<c­<∞
Σan converges ⟺ Σbn converges
Σan diverges ⟺ Σbn diverges
Inconc­lusive if c=0, c=∞, or c DNE
Ratio Test
Σan, usually with n! terms, product terms, or (an)n
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 (an)n
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 Classi­fic­ation
Σan
Absolutely converges if Σ|an| converges
Diverges if Σan diverges
Condit­ionally converges if Σ|an| diverges, but Σan converges
For the series listed, assume each series to be an infinite series starting at n=k: Σ∞n=k=Σ as previously defined
If a test is inconc­lusive, 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

Special Series

Series
Summation Form
First five terms
Radius & Interval of Conver­gence
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+...
(a-R, a+R), [a-R, a+R), (a-R, a+R], or [a-R, a+R]
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!+...
|x-a|<R
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!+...
|x|<R
1/(1-x)
Σxn
1+x+x2+x3+x4+...
R=1, I=(-1,1)
ex
Σxn/n!
1+x+x2/2!+x3/3!+x4/4!+...
R=∞, I=(-∞,∞)
ln(1+x)
Σ(-1)nxn+1/(n+1)
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)!, where M is a constant such that |f(n+1)(x)|≤M 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 as previously defined
The Radius of Conver­gence, R, is typically found by using the Ratio Test or Root Test
(kn) is the "­bin­omial coeffi­cie­nt" (read as "k choose n"). (kn) = k!/(n!­(k-n)!)
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: favg=1/(b-a)∫ab f(x)dx

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)

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

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
  

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"

Choosing the "­dv" term depends on what will simplify the integral the best, while being relatively simple to integrate

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

Trigon­ometric Integrals

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
  

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->∞ ∫xt f(x)dx
Conver­gence of ∫f(x)dx
lim t->±∞ f(x)dx=L, where L is a constant
Divergence of ∫f(x)dx
lim t->±∞ f(x)dx=±∞ or DNE
Improper Integrals are integrals with bounds at infinity (Type 1) or at least one discon­tinuity on the integrated region (Type 2)

Conic Sections

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 Parameters r and 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
  

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(x1))
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, that is, n=2k for some integer k

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

Metadata

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  • Published: 28th May, 2023
  • Last Updated: 10th April, 2025

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

CROSSANT CROSSANT, 06:03 18 Dec 23

Hey that's me

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