If the area of the region bounded by the curves x = 2a^3 y^2 + a^2 and x = y^2 a (where a > 0 ) is 25 - 50 3 , then the value of a is:

Step-by-step solution

To find the points of intersection, we equate the two expressions for x : 2a^3 y^2 + a^2 = y^2 a y^4 + a^2y^2 - 2a^4 = 0 (y^2 + 2a^2)(y^2 - a^2) = 0 y = a (since y^2 0 and a > 0 ) The area of the bounded region is: A = _ -a ^ a ( 2a^3 y^2 + a^2 - y^2 a ) dy Using the property of even functions: A = 2 ₀^ a ( 2a^3 y^2 + a^2 - y^2 a ) dy = 2 [ 2a^3 ( 1 a ⁻¹ ( y a ) ) - y^3 3a ]₀^ a = 2 [ 2a^2 ⁻¹(1) - a^2 3 ] = 2 ( 2a^2 ( 4 ) - a^2 3 ) = a^2 ( - 2 3 ) Given that the area is 25 - 50 3 = 25 ( - 2 3 ) , we have: a^2 ( - 2