Circles and ellipses

The circle x² + y² = a² is symmetric about both axes, so its area is 4 times the area of the quarter in the first quadrant. There y = √(a² − x²) is taken positive.

Area of the circle = 4∫₀ᵃ √(a² − x²) dx = 4[(x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a)] from 0 to a = 4 × (a²/2)(π/2) = πa². Horizontal strips, with x = √(a² − y²), give the same result.

The ellipse x²/a² + y²/b² = 1 gives y = (b/a)√(a² − x²) in the first quadrant. Its area is 4(b/a)∫₀ᵃ √(a² − x²) dx = 4(b/a)(πa²/4) = πab.

An ellipse is a circle stretched by the factor b/a in one direction, so every vertical strip, and hence the area, is scaled by b/a: πa² × (b/a) = πab. With a = b this is the circle again.

Worked example: x² + y² = 16 has area 16π; its quarter in the first quadrant has area 4π.

Worked example: x²/25 + y²/9 = 1 has a = 5 and b = 3, so its area is 15π.

Worked example: the part of x² + y² = 4 to the right of x = 1 has area 2∫₁² √(4 − x²) dx = 4π/3 − √3.