If the area of the region enclosed by the parabolas y = 2x^2 and y = c - 2x^2 (where c > 0 ) is 18 square units, then the value of c is

Step-by-step solution

The given parabolas are y = 2x^2 and y = c - 2x^2 . To find the points of intersection, we equate the two expressions for y : 2x^2 = c - 2x^2 4x^2 = c x = c 2 The region is symmetric about the y -axis. The upper boundary is y = c - 2x^2 and the lower boundary is y = 2x^2 . The area A of the enclosed region is given by: A = _ - c /2 ^ c /2 ( (c - 2x^2) - 2x^2 ) dx A = 2 ₀^ c /2 (c - 4x^2) dx Integrating, we get: A = 2 [ cx - 4x^3 3 ]₀^ c /2 A = 2 ( c ( c 2 ) - 4 3 ( c c 8 ) ) A = 2 ( c c 2 - c c 6 ) A = 2 ( 3c c - c