AP Calc BC Integrals and Formulas

Area between two polar curves

• R is outer

• r is inner

• Limits of integration (alpha & beta) are the intersections of R and r, so solve that first

Area between two curves

Volume of disk or washer (curve revolved around x-axis

• If revolved around different horizontal line y=#, change every x to x-#

• If revolving around y-axis or vertical line, first solve for x and then same idea^^^ if around x=#, change every y to y-#

Arc length (NOT parametric)

Parametric arc length

Parametric first derivative

Parametric second derivative

Average value of f (Mean Value Theorem)

Total Distance vs Displacement formulas

Speed (magnitude of vector)

Integration by parts

Just use if u-substitution doesn’t work

Partial fractions

Use for integration if u-substitution and integration by parts don’t work

Formal definition of an integral

• n = total # of rectangle subintervals

• Δx = width of subintervals

• i = i^th subinterval of Right Riemann

For Left Riemann, change inside to a+(i-1)Δx or change start to i=0

Trapezoid Riemann sum

Trapezoid concavity rules

Local/relative extrema tests

Extreme Value Theorem

Absolute extrema must be at critical values or endpoints of interval!!!!!!!!!!

Find crit vals with f’(x)=0 then plug those^^^ x values into f(x) to find absolute min/max

Logistic growth formulas

Semicircle formula

If you have to take the integral of this, just do ½ Area of circle (or integral halfway through it is ¼ Area of circle)

0<a<1

a>0

Differentiability Requirements

• f is continuous

• Limits from each side are equal

• No cusps / sharp edges

Differentiable implies continuous!!!!! (But not other way around)

Exponent rules

Limit definitions of vertical/horizontal asymptotes

Even/odd function stuff