Parabolas:

• $y^2$ = 4ax, latus rectum x = a: Area = 8$a^2$/3
• $y^2$ = 4ax, line y = mx: Area = 8a^$\frac{2}{3m^3}$
• $y^2$ = 4ax, $x^2$ = 4by: Area = 16ab/3
• y = $x^2$, y = x (0 to 1): Area = 1/6
• y = $x^2$, y = sqrt(x) (0 to 1): Area = 1/3
• Quadratic between roots: |a|(beta-alpha)^3/6
• y = $x^2$, y = a: Area = 4a^$\frac{3/2}{3}$

Conics:

• Circle x^{2+y}^2 = $r^2$: Area = pi*$r^2$
• Ellipse $x^2$/$a^2$ + $y^2$/$b^2$ = 1: Area = piab
• Circular segment (angle theta, radius r): ($r^2$/2)(theta - sin(theta))

Trigonometric:

• One arch of sin(x): Area = 2
• cos(x) from 0 to pi/2: Area = 1
• |sin(x)| over n arches: Area = 2n

Special Regions:

• |x/a| + |y/b| = 1: Area = 2ab
• |x-h| + |y-k| = c: Area = 2$c^2$ (diamond centered at (h,k))

Polar:

• Cardioid r = a(1+cos(theta)): Area = 3pi$a^2$/2
• Rose r = acos(ntheta): one petal = pia^$\frac{2}{4n}$ for n even, pia^$\frac{2}{2n}$ for n odd; total petals = 2n (even) or n (odd)