__Question:__
Find

such that the area of the region enclosed by the parabolas

and

is

.

My problem probably lies with the limits of integration. When I get x^2 = c^2 x = c, I assumed the limits of integration would be -c and c but I can't justify it not to mention the answer is wrong.

Can someone please help me?

Any help would be greatly appreciated!

Thanks in advance!

If you are still wondering about why the limits of integration are \(\displaystyle -c<x<c\), you of course know we set the two expressions equal:

\(\displaystyle x^2-c^2=c^2-x^2\)

\(\displaystyle \implies 2x^2 = 2c^2\)

\(\displaystyle \implies x^2 = c^2\)

\(\displaystyle \implies \sqrt{x^2} = \sqrt{c^2}\)

Now, here is where most people make a mistake (or try to take shortcuts). What is the square root of \(\displaystyle x^2\) ? It's \(\displaystyle |x|\), NOT \(\displaystyle x\). So our next step will be:

\(\displaystyle \implies |x| = c\)

\(\displaystyle \implies x = \pm c\)

By the way, the reason we can say that \(\displaystyle \sqrt{c^2}=c\) is because the problem states that \(\displaystyle c>0\).