Find the values of c such that the area of the region bounded by the parabolas
y = 16x^2 − c^2 and y = c^2 − 16x^2
is 144

For there to be a region bounded by the two parabolas, you first need to find some conditions on . The two parabolas must intersect each other twice, so you need two solutions to



You have



which means you only need to require that . With that, the area of any such bounded region would be given by the integral



since for all . Now,



by symmetry across the y-axis. Integrating yields







Since , you have .

The answer maybbe c