Answer to Question #348509 in Calculus for Migz

Question

The loop of the curve has an equation of y² = x (1 - x)^2. Find the

area enclosed by the loop of the curve.

Answer 1

Simplify the function: $y=\pm \sqrt{x}(1-x)$.

Limits of integration: $x=0$, $x=1$.

$S=2\int\limits_0^1\sqrt{x}(1-x)dx=2\int\limits_0^1(\sqrt{x}-x\sqrt{x})dx=2\int\limits_0^1(x^{1/2}-x^{3/2})dx=$

$=2(\frac{x^{3/2}}{3/2}-\frac{x^{5/2}}{5/2} )|_0^1=2(\frac{2x^{3/2}}{3}-\frac{2x^{5/2}}{5} )|_0^1=2(\frac{2}{3}-\frac{2}{5})=\frac{8}{15}$.