AP Calculus BC

shell method: revolved around y-axis

V=2π∫p(x)h(x)dx

shell method: revolved around x-axis

V=2π∫p(y)h(y)dy

arc length

L=∫√(1+(f’(x))²dx

surface area: revolved around x-axis

S=2π∫f(x)√(1+(f’(x))²dx

surface area: revolved around y-axis

S=2π∫x√(1+(f’(x))²dx

work

W=∫F(x)dx

Hooke’s Law

F=kx

Newton’s Law

F=C/x²

Boyle’s Law

P=k/V

mass

m=force/acceleration

moment

moment=mass∙distance

moment about the origin: one-dimensional system

M₀=m₁x₁+…+m_nx_n

moment about the y-axis: two-dimensional system

M_y=m₁x₁+…+m_nx_n

moment about the x-axis: two-dimensional system

M_x=m₁y₁+…+m_ny_n

center of mass: two-dimensional system

(x̄=M_y/m,ȳ=M_y/m) where m=m₁+…+m_n

moment about the y-axis: planar lamina

M_y=∫x(f(x)-g(x))dx

moment about the x-axis: planar lamina

M_x=∫((f(x)+g(x))/2)(f(x)-g(x))dx

center of mass: planar lamina

(x̄=M_y/m,ȳ=M_y/m) where m=∫(f(x)-g(x))gx

pressure

P=density∙depth

fluid force on a horizontal surface

F=pressure∙area

force exerted by a fluid

F=∫h(y)L(y)dy

∫(tanu)du

-ln|cosu|+C

∫(cotu)du

ln|sinu|+C

∫(secu)du

ln|secu+tanu|+C

∫(cscu)du

-ln|cscu+cotu|+C

∫(sec²u)du

tanu+C

∫(csc²u)du

-cotu+C

∫(secu∙tanu)du

secu+C

∫(cscu∙cotu)du

-cscu+C

∫du/√(1-x²)

arcsin(u/a)+C

∫du/(a²+u²)

(1/a)arctan(u/a)+C

integration by parts theorem

∫u∙dv=u∙v-∫v∙du

guidelines for choosing “u” in integration by parts

Logarithm, Inverse trigonometric, Algebraic, Trigonometric, Exponential

Pythagorean identity

sin²x+cos²x=1

power reduction formula for sin²x

sin²x=(1-cos2x)/2

power reduction formula for cos²x

cos²x=(1+cos2x)/2

sine product with different angles

sin(mx)∙sin(nx)=1/2(cos((m-n)x)-cos(m+n)x))

sine-cosine product with different angles

sin(mx)∙cos(nx)=1/2(sin((m-n)x)+sin(m+n)x))

cosine product with different angles

cos(mx)∙cos(nx)=1/2(cos((m-n)x)+cos(m+n)x))

trigonometric substitution for √(a²-x²)

x=asinθ

trigonometric substitution for √(a²+x²)

x=atanθ

trigonometric substitution for √(x²-a²)

x=asecθ

logistic differential equation form

dy/dx=ky(1-y/L)

logistic growth equation form

y=L/(1+be^-kt)

At what point does a logistic growth curve increase the fastest?

when the curve reaches half its carrying capacity (L)

What is the parametric form of the derivative?

dy/dx=(dy/dt)/(dx/dt), dx/dt≠0

When does a smooth curve have a horizontal tangent line?

at t where dy/dt=0, but dx/dt≠0

When does a smooth curve have a vertical tangent line?

at t where dx/dt=0, but dy/dt≠0

What two formulas do you use to convert polar coordinates to rectangular coordinates?

x=r∙cosθ and y=r∙sinθ

What three formulas do you use to convert rectangular coordinates to polar coordinates?

r=√(x²+y²), θ=arctan(y/x) when x>0, and θ=arctan(y/x)+π when x<0

slope in polar form

dy/dx=(f(θ)cosθ+f’(θ)sinθ)/(-f(θ)sinθ+f’(θ)cosθ)

area in polar coordinates

A=1/2∫(f(θ))² dθ

arc length in polar coordinates

L=∫√(r²+(dr/dθ)²) dθ

general expression for a geometric sequence

a₁∙r^n-1

sum of a convergent geometric series

S=(first term of the series)/(1-the common ratio)

limit of the n^th term of a convergent series

if the sum of a_n converges, then (lim(n→∞))(a_n)=0

n^th term for divergence

if (lim(n→∞))(a_n)≠0, then the sum of a_n diverges