Finding area between two curves using double integral Find the area between two curves:


*

*$x \ge 0$

*$(x^2+y^2)^2=x^2-y^2$
using double integrals.

Here's the area:

Solve $(x^2 + y^2)^2 = x^2 − y^2$ for $y(x)$ and get four solutions, restrict to positive $x$ to find:
$$y(x) = \pm \sqrt{2} \sqrt{-1 - 2 x^2 + \sqrt{1 + 8 x^2}}$$
so:
$$\int\limits_{x=0}^1 2 \sqrt{2} \sqrt{-1 - 2 x^2 + \sqrt{1 + 8 x^2}}\ dx = \frac{1}{2}$$
(I have no idea why there might be a need for a double integral.)
