The curve with equation $\displaystyle a^2 y^2 = x^2 (a^2 - x^2)$ has two loops.
Show, by integration, that the area enclosed by a loop is $\displaystyle \frac {2}{3} a^2$.
Please help me to do this.
solving the above equation for y we get
$\displaystyle y=\frac{x\sqrt{a^2-x^2}}{a}$
$\displaystyle \int_0^{a}\frac{x\sqrt{a^2-x^2}}{a}dx$
is half of the area of one loop. (Draw a graph to see)
$\displaystyle 2\int_0^{a}\frac{x\sqrt{a^2-x^2}}{a}dx=$
is the area of one loop. just let $\displaystyle u=a^2-x^2$
good luck.