Application of Definite Integration — Formula Sheet

Parabolas

y² = 4ax (opens right), y² = -4ax (opens left), x² = 4by (opens up), x² = -4by (opens down)

Ellipse

x²/a² + y²/b² = 1 where a and b are semi-major and semi-minor axes

Area between curve y = f(x), x-axis, and vertical lines x = a, x = b is ∫[a to b] f(x) dx

Area between y = f(x) and y = g(x) is ∫[a to b] |f(x) - g(x)| dx

Area of circle x² + y² = r² is πr²

General form

x²/a² + y²/b² = 1.

For curve y = f(x) between x = a and x = b: A = ∫[a to b] f(x) dx.

For curve x = g(y) between y = c and y = d: A = ∫[c to d] g(y) dy.

Take absolute value

|∫[a to b] f(x) dx|.