AP Calculus 3rd Quarter

Average Value

1/ b

(b-a) ∫ f(x) dx

a

Value of a Function

Initial + Accumulation

Displacement

∫ v(t)

Distance

∫ abs value (v(t))

Position

s(0) + ∫ v(t)

Net Change

Starting amount at time a + ∫(Rate in) dt - ∫(Rate out) dt

Area between two curves in terms of x

∫ (Upper - lower) dx

Area between two curves in terms of y

∫ (Right - left) dy

Volume (Disk Around X-Axis)

V= π∫R² dx or V= π∫f(x)² dx

Volume of Revolution (Washer - Around X-Axis)

V= π∫(R²-r²) dx or V= π∫(f(x)²-g(x)²)

R

Outer Curve

r

Inner Curve

Disk Around Other Axis

V= π∫(f(x)-L)² dx

Washer Around Other Axis

V= π∫(f(x)-L)²-(g(x)-L)² dx

Square Area

A = (side)²

Rectangle Area

A= base x height

Semi-Circle Radius

A=1/2πr²

Triangle Area

A=1/2bh

General Volume of a Cross Section

V= ∫ Area dx

Square Volume Cross Section

V = ∫ side² dx

Side

Upper-lower