x^2 / a^2 + y^2 / b^2 = 1
(bx)^2 + (ay)^2 = (ab)^2
y=+/- sqrt [b^2 - (bx/a)^2]
area =2*∫ y dx ( upper limit = a , lower limit = -a)
=2*∫ sqrt [b^2 - (bx/a)^2] dx
=2b ∫ sqrt [1-(x/a)^2] dx using substitution, x=a sin Θ
=2b ∫ sqrt [1-(sinΘ)^2] d(a sinΘ) (upper limit = π/2 , lower limit = -π/2)
=2ab ∫ (cosΘ)^2 dΘ
=ab∫ (1+cos2Θ)dΘ
=ab [Θ+(sin2Θ)/2](upper limit = π/2 , lower limit = -π/2)
=ab(π/2 - (-π/2))
=πab
as a reference, circle is a special case of an ellipse with area= πr^2 , since a=b=r in a circle.
