The area of the region bounded by the ellipse x^2 a^2 + y^2 b^2 =1 is

Step-by-step solution

The equation of the ellipse is x^2 a^2 + y^2 b^2 = 1 . Solving for y , we get y = b a a^2 - x^2 . The area A of the ellipse is given by 4 ₀^ a y , dx = 4 ₀^ a b a a^2 - x^2 , dx . A = 4b a ₀^ a a^2 - x^2 , dx . Using the standard integral a^2 - x^2 , dx = x 2 a^2 - x^2 + a^2 2 ⁻¹ ( x a ) , we evaluate the definite integral: A = 4b a [ x 2 a^2 - x^2 + a^2 2 ⁻¹ ( x a ) ]₀^ a . A = 4b a ( 0 + a^2 2 ⁻¹(1) - (0 + 0) ) = 4b a a^2 2 2 . A = a b . Answer: a b sq units