# Calc1.75

Find the area under the curve $f(x)=x^{2}$ on the interval $[-1,1]$.
$A=\int _{{-1}}^{{1}}x^{2}\,dx\,$
The first step you can use to simplify this integral a tiny bit is to recognize that $x^{2}$ is an even function. An even function is one for which it is true that $f(-x)=f(x)$ for all x. Since this is true, it has a symmetry about $x=0$. Thus, the value of this integral on $[-1,1]$ is simply twice that of the integral on $[0,1]$. In some cases, this is a great help; in others, not so much.
$A=\int _{{-1}}^{{1}}x^{2}\,dx=2\int _{{0}}^{{1}}x^{2}\,dx={\frac {2}{3}}x^{3}{\bigg |}_{{0}}^{{1}}={\frac {2}{3}}\,$
Thus, the area between the function and the x-axis, on the interval $[-1,1]$ is ${\frac {2}{3}}$.