Definite Integrals — Area Under the Curve

∫ from a to b of f(x)dx = net signed area between f(x) and the x-axis from x=a to x=b.

Fundamental Theorem of Calculus (Part 1):

∫ from a to b of f(x)dx = F(b) - F(a), where F is any antiderivative of f.

Net vs total area:

Net area: area above x-axis minus area below.

Total area: ∫|f(x)|dx — take absolute value first.

Properties:

∫ from a to a = 0

∫ from a to b = -∫ from b to a

∫ from a to b + ∫ from b to c = ∫ from a to c

∫ from a to b [cf(x)]dx = c · ∫ from a to b f(x)dx

Riemann sums approximate the integral: left, right, midpoint, trapezoidal. The AP exam tests all four.

Reference:

Wikipedia: Integral

https://en.wikipedia.org/wiki/Integral