Question

Find the area of the ellipse `x^2/1 + y^2/4` = 1, in first quadrant

Answer 1

Given equation of the ellipse is `x^2/1 + y^2/4` = 1.

∴ `y^2/4` = 1 − x²

∴ y² = 4(1 – x²)

∴ y = `+-  2sqrt(1 - x^2)`

∴ y = `2sqrt(1 - x^2)`   ......[∵ In first quadrant, y > 0]

∴ Required area

=  `int_0^1 y  "d"x`

= `int_0^1 2sqrt(1 - x^2)  "d"x`

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

= `2[0 + 1/2 sin^-1 (1) - 0]`

= `2[1/2 (pi/2)]`

= `pi/2` sq.units