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