AP Calculus AB - Unit 6: Stuff to know

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When cross sections to base are squares

Integrand is squared (x)²

When cross sections to base are semicircles

Function is multiplied by π/2, Integrand is divided by two and squared, (x/2)²

When cross sections to base are isosceles right triangles

Function is multiplied by 1/4, Integrand is squared

When cross sections to base are equilateral triangles

Function is multiplied by (√3)/4, Integrand is squared

When cross sections are perpendicular to x-axis

Integral goes from a-b on the X-AXIS

When cross sections are perpendicular to y-axis

Integral goes from a-b on the Y-AXIS

When cross sections are circles (disks)

Function is multiplied by π, Integrand is squared

Half-Life Formula (1)

A(t) = A₀(1/2)ᵗ/ʰ

A = amount

A₀ = initial amount

t = time

h = half-life

Newton Cooling Formula (Derivative form)

dy/dt=k(y-Tₐ)
y = Temperature
k = Cooling Constant
t = time
Tₐ = Ambient Temperature

Exponential Growth/Decay Formula

y₁ = y₀eᵏᵗ

y₁ = amount

y₀ = initial amount

k = rate constant

t = time

Newton Cooling Formula

T(t) = Tₛ + (T₀ - Tₛ)e⁻ᵏᵗ

T = temperature

Tₛ = surrounding temperature

T₀ = initial temperature

k = rate constant

t = time

Trapezoidal Sum

Tₙ = ((b-a)/2n)*(y₀ + 2y₁ + 2y₂ + ……. + yₙ₋₁ + yₙ)

Newton’s Law of Cooling (Separated)

ln(T - Tₐ) = kt + C

T = temperature

Tₐ = surrounding temperature

k = rate constant

t = time

C = constant (determined by calculation)

General formula for the volume of a disk

V = pi integral a-b(r²)dx

r = radius, height of rectangular strip of the region

General formula for the volume of a washer

V = pi integral a-b(R²-r²)dx

R = outer radius, distance to the top of rectangular strip

r = inner radius, distance to the bottom of the strip

Annual Compounding Formula

A(t) = A₀(1+r)ᵗ

r = rate of interest

A₀ = initial amount

t = time

Continuous Compounding Formula

A(t) = A₀(1+(r/n))ⁿᵗ OR A(t) = A₀*eʳᵗ

n = number of times original amount is compounded

r = rate of interest

A₀ = initial amount

t = time