AP Calc BC Midterm Equations

Area

∫ f(x) - g(x) dx

Volume

π ∫ (f(x))² dx where f(x) is radius

Washer method volume (x axis)

π ∫ (f(x))²- (g(x))² dx

Shell method volume (y axis)

2π ∫ x(f(x) - g(x)) dx where x is the radius and f(x) and g(x) are the height functions.

Coefficient of semicircle

π/8

Coefficient of isosceles right triangle where triangle is sitting up like how we usually picture a right triangle

1/2

Coefficient for isosceles right triangle where it looks like its the roof of a house and right angle is on the top angle and the equal sides are the two sides of the roof (not hypotenuse)

1/4

coefficient of square

1

Coefficient of rectangle

K but there are two ways for rectangle K ∫ (f(x) - g(x)) dx OR ∫ h(x) (f(x) - g(x))dx

Arc length

∫ √(1 + (f’(x))²) dx

Surface Area

2π ∫ r(x) √(1 + (f’(x))²) dx

Area of Elipse

4a/b ∫ √(a² - x²) dx (bounds from 0 to a)

Partial fractions

separates denominator into fractions

Integration by parts

∫ uv’ = uv - ∫vu’

Trig sub: √(a²-x²)

x = a sin(θ)

Trig sub: √(a²+ x²)

x = a tan(θ)

Trig sub: √(x² - a²)

x = a sec(θ)

Tabular

∫ (polynomial)(exponential or sin or cos)

Indefinite forms of L’Hopital’s Rule

(0/0), (∞/∞), (0*∞), (∞ - ∞), (0⁰), (∞^{^0}), (1∞)

Improper integrals

Can be something with bounds to negative or positive infinity which will end up with a horizontal asymptote at y=0 OR can be something with a discontinuity in function in the integral bounds which will mean it will have a vertical asymptote

Separable differential equations

Separate (x and y), integrate, celebrate

Logistics P equation

P = (L/(1+Be^-kt))

Logistics P’ equations

P’ = KP(L-P) OR P’ = K(1-(P/L))

Euler’s method

In calc put →A then →B then B + Δx(A+B) →B and then finally A + Δx →A

Parametric slope (derivative)

(dy/dx) = (dy/dt)/(dx/dt)

Parametric second derivative

[(d/dt)(dy/dx)]/(dx/dt)

Parametric Arc Length

∫ √( (x’)²+(y’)²) dt (bounds are t₁ to t₂)

Parametric Surface Area

2π ∫ r √( (x’)²+(y’)²) dt

Parametric integral

∫ y(t) (dx/dt)dt

Parametric volume

π ∫ (y(t))²(dx/dt)dt

Cartesian to Polar

θ = arctan(y/x)

Polar to Cartesian

x = r cos(θ) and y = r sin(θ)

Polar Slope in plane (derivative)

(dy/dx) = (dy/dt)/(dx/dt)

Polar Arc Length

∫ √((r’)²+r²) dθ

Polar Area

½ ∫ r² dθ

Polar Area between 2 curves/shapes

½ ∫ (f(θ))² - (g(θ))² dθ