Calculus 2 Final Exam Review

McLaurin Series for 1/(1-x)

∑xⁿ=1+x+x²+x³+x⁴+...

R=1

McLaurin Series for e^x

∑xⁿ/n!=1+x/1!+x²/2!+x³/3!+...

R=infinity

McLaurin Series for sinx

∑(-1)ⁿ(x²ⁿ⁺¹)/(2n+1)!=x-x³/3!+x⁵/5!-x⁷/7!+...

R=infinity

McLaurin Series for cosx

∑(-1)ⁿ(x²ⁿ)/(2n)!=1-x²/2!+x⁴/4!-x⁶/6!+...

R=infinity

Ratio Test

L=1 (nothing) L<1 (converges) L>1 (diverges)

The Alternating Series Test

Integral Test

Geometric Series

Divergence Test

P-Series Test

SA Rotation about x-axis

Uses y

SA Rotating about y-axis

Uses x

Alternating Series Estimation

The Comparison Test

(i) If ∑bn is convergent and an≤bn for all n, then ∑ an is also convergent.

(ii) If ∑bn is divergent and an≥bn for all n, then ∑ an is also divergent

.

Washer Method

pi∫(outer radius)²-(inner radius)²

(x-axis)

Shell Method

2pi∫rh

(y-axis)

Integration by Parts

uv-∫vdu

Arc Length Formula

∫√(1+(dy/dx)^2)

∫lnu

ulnu-u

sin2x

2sinxcosx

sin²x

(1-cos2x)/2

cos²x

(1+cos2x)/2

∫secxdx

ln|secx+tanx|

Derivative of secx

secxtanx

Derivative of tanx

sec²x

Derivative of cscx

-cscxcotx

Derivative of cotx

-csc²x

Length of Polar Curve

∫√(r²+(dr/d∅)²)d∅

derivative of sinx

cosx

derivative of cosx

-sinx

integral of sinx

-cosx

integral of cosx

sinx

f average

(1/B-A)∫f(x)dx

Area

b

∫[f(x)-g(x)]dx

a

√(a²-x²)

x=aSinθ

√(x²-a²)

x=aSecθ

√(a²+x²)

x=aTanθ

Integral Test Remainder Estimate

Limit of arctan(n)

n-->∞

π/2