Find the area of ellipse `x^(2)/1 + y^(2)/4 = 1.`

Explanation

By symmetry of ellipse, required area of the ellipse is 4 times the area of region OPQO.
` (##GRK_10Y_SP_HSC_MAT_MAR_18_E02_006_S01.png" width="80%">
Equation of ellipse is
` x^(2)/1 + y^(2)/4 = 1 ` ….(1)
` :." " a = 1, b = 2`
` :. ` Limits of integration are x = 0 and x = 1.
From equation (1), we get
` 4x^(2) + y^(2) = 4`
` rArr y^(2) = 4 - 4x^(2)`
` rArr y^(2) = 4 (1 - x^(2))`
` rArr y = pm 2 (sqrt(1-x^(2))`
` rArr y = 2 sqrt(1-x^(2))`
` :' ` In first quadrant, ` y gt 0`
` :. ` Area of the ellipse is
A = 4 ( Area of region, OPQO)
` = 4 int_(0)^(1) y * dx `
` = 4 int _(0)^(1) 2 sqrt(1 - x^(2)) * dx `
` = 8 int_(0)^(1) sqrt(1 - x^(2)) * dx `
` = 8 [1/2 x * sqrt(1 -x^(2))+ 1/2 sin^(-1) x]_(0)^(1)`
` = 8 [0 + 1/2 sin^(-1) 1 -0 -1/2 sin^(-1) 0]`
` = 8 [1/2 * pi/2 - 1/2 * 0]`
` = 8 xx pi/4 = 2 pi ` sq. units .