Integration problem

**geton** · April 21st 2008, 09:55 PM

The curve with equation $a^2 y^2 = x^2 (a^2 - x^2)$ has two loops.
Show, by integration, that the area enclosed by a loop is $\frac {2}{3} a^2$ .

Please help me to do this.

**TheEmptySet** · April 21st 2008, 10:11 PM

Originally Posted by geton

The curve with equation $a^2 y^2 = x^2 (a^2 - x^2)$ has two loops.
Show, by integration, that the area enclosed by a loop is $\frac {2}{3} a^2$ .

Please help me to do this.

solving the above equation for y we get

$y=\frac{x\sqrt{a^2-x^2}}{a}$

$\int_0^{a}\frac{x\sqrt{a^2-x^2}}{a}dx$

is half of the area of one loop. (Draw a graph to see)

$2\int_0^{a}\frac{x\sqrt{a^2-x^2}}{a}dx=$

is the area of one loop. just let $u=a^2-x^2$

good luck.

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