Question

Find the values of `c` such that the area...?

Find the values of `c` such that the area of the region bounded by the parabolas `y=x^2-c^2` and `y=c^2-x^2` is 576.

Answer 1

`c =6`

Explanation:

The two curves are:

`y_1(x) = x^2-c^2`

and

`y_2(x) = c^2-x^2`

We can note that for every `x` we have: `y_1(x) = -y_2(x)` so the two parabolas are symmetric with respect to the `x` axis.The two curves thus intercept when `y_1(x) = y_2(x) = 0`, that is for `x=+-c`

Given the symmetry, the area bounded by the two parabolas is twice the area bounded by either parabola and the `x` axis.

If we choose `y_2(x) = c^2-x^2`, which is positive in the interval, we thus have:

`A = 2 int_(-c)^c ( c^2-x^2)dx = 2 [c^2x-x^3/3]_(-c)^c = 8/3c^3`

and posing

`8/3c^3 = 576`

we get:

`c = root(3)((3 xx 576)/8) = 6`