Question

Compute the area of the loop of the curve $y^2=x^2\left[\right(1+x\left)\right(1-x\left)\right]$

Answer 1

When we plot the graph of $y^2=x^2\left[\right(1+x\left)\right(1-x\left)\right]$

we get an infinity symbol.

${\int }_0^1{x}^2\left[\right(1+x\left)\right(1-x\left)\right]dx$

$={\int }_0^1{x}^2[1-x^2]dx$

$={\int }_0^1(x^2-x^4)dx$

$={[\frac{x^3}{3}-\frac{x^5}{5}]}_0^1$

$=\frac{1}{3}-\frac{1}{5}$

$=\frac{2}{15}sq.units$