Key Formulas for Area Under Curves

Basic area: A = integral from a to b of f(x) dx (signed), A = integral of |f(x)| dx (geometric)
Between curves: A = integral from a to b of [f(x) - g(x)] dx, where f(x) >= g(x)
Horizontal strips: A = integral from c to d of [ $x_{right}$ (y) - $x_{left}$ (y)] dy
Parametric: A = |integral from t1 to t2 of y(t) * x'(t) dt|
Polar: A = $\frac{1}{2}$ * integral from theta1 to theta2 of $r^2$ d(theta)
Ellipse: A = piab
Parabola + latus rectum ( $y^2$ = 4ax, x = a): A = 8 $a^2$ /3
Parabola + line ( $y^2$ = 4ax, y = mx): A = 8a^ $\frac{2}{3m^3}$
Two parabolas ( $y^2$ = 4ax, $x^2$ = 4by): A = 16ab/3
Quadratic + x-axis (roots alpha, beta): A = |a|(beta - alpha)^3/6
y = $x^2$ and y = x: A = 1/6
|x/a| + |y/b| = 1: A = 2ab
Even function: integral from -a to a = 2 * integral from 0 to a
Odd function: integral from -a to a = 0
One arch of sin(x): A = 2
Cardioid r = a(1+cos(theta)): A = 3pi $a^2$ /2
Complement identity: integral of f + integral of f^(-1) = bd - ac
Area between y = $x^2$ and y = a: A = 4a^ $\frac{3/2}{3}$
Chord of parabola y = $x^2$ from (a, $a^2$ ) to (b, $b^2$ ): A = |b-a|^3/6
Tangent to ellipse and axes: min area = ab