Question

Find the area of the ellipse `x^2/36 + y^2/64` = 1, using integration

Answer 1

By the symmetry of the ellipse, required area of the ellipse is 4 times the area of the region OPQO.

For the region OPQO, the limits of integration are x = 0 and x = 6.

Given equation of the ellipse is `x^2/36 + y^2/64` = 1

∴ `y^2/64 = 1 - x^2/36`

∴ y² = `64(1 - x^2/36)` 

∴ y² = `64/36(36 - x^2)`

∴ y = `+-  8/6 sqrt(36 - x^2)`

∴ y = `4/3 sqrt(36 - x^2)`  ......[∵ In first quadrant, y > 0]

∴ Required area = 4(area of the region OPQO)

= `4int_0^6 y  "d"x`

= `4 int_0^6 4/3 sqrt(36 - x^2)  "d"x`

= `16/3[x/2 sqrt(36 - x^2) + 36/2 sin^-1 (x/6)]_0^6`

= `16/3[6/2 sqrt(36 - 36) + 36/2 sin^-1 (1) - {0 +36/2 sin^-1 (0)}]`

= `16/3(0 + 36/2* pi/2 - 0)`

= 48π sq.units