Calculus 2 Formulas

Area between curves

∫ (g(x) - f(x)) dx

Volume of solids with washer method

∫pi * (R^2 - r^2) dx

R - r2

r - r1

Volume of solids with disk method

∫pi * (f(x))^2 dx

Integration by parts

∫ u dv = uv - ∫ v du

What do you use for this trig substitution: √(x^2 + a^2)

x = atan(θ)

dx = asec^2(θ) dθ

-pi/2 < θ < pi/2

What do you use for this trig substitution: √(x^2 - a^2)

x = asec(θ)

dx = asec(θ) tan(θ) dθ

θ in QI and QIII

What do you use for this trig substitution: √(a^2 - x^2)

x = asin(θ)

dx = acos(θ) dθ

-pi/2 < θ < pi/2

P-series rules

1/n^p converges if p>1

diverges if p<1

Arc length

∫ √( 1 + (f'(x))^2 ) dx

Surface area

2pi ∫ f(x) √( 1 + (f'(x))^2 ) dx

What are the rules for the integral test?

When x >= 1:

Is it continuous?

Is it decreasing?

Is f(x) > 0 ?

Integral test

(b->∞)lim ∫ ...

What order should you do if you are finding absolute convergence?

Limit, direct, then integral

Direct comparison

0 < An < Bn

Parabola equation

(x-h)^2 = 4c(y-k)

Ellipse equation

(x-h)/a^2 + (y-k)/b^2 = 1

a > b

Hyperbola equation

(y-k)^2/a^2 - (x-h)^2/b^2 = 1

a^2 + b^2 = c^2

a is always on the positive fraction

a : distance from center to vertex. 0

Power series

Converges when x = a, R = 0

Converges for all x, R = 0

Converges when | x-a | < R <=> -R < x-a < R

Maclaurin series for f(x) = ...

(Sum from 0 to infinity) ( ... )^n / n!

Arc length along x = x(t) , y = y(t), a <= t <= b

s = ∫ √( x'(t))^2 + (y'(t))^2 ) dt

Convert to polar equations (x,y)

r^2 = x^2 + y^2

tanθ = y/x

Convert to Cartesian / rectangular equations (r , θ)

x = rcosθ

y = rsinθ

Finding tangent line of polar graph

( f(θ) cosθ + f'(θ) sinθ ) / -( f(θ) sinθ + f'(θ) sinθ

Area of polar graphs

1/2 ∫ [f(θ)]^2 dθ

Arc length of polar equations

∫ √( (f(θ))^2 + (f'(θ))^2 ) dθ