Parabolas:

• y^2 = 4ax, latus rectum x = a: Area = 8a^2/3
• y^2 = 4ax, line y = mx: Area = 8a^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^(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 = 2c^2 (diamond centered at (h,k))

Polar:

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