Intergral of a symmetric function

Problem

Demonstrate that the graph of the function \(g:x\mapsto x^2\cdot \sin x\) defined over \(\mathbb{R}\) is point symmetric with respect to the coordinate origin, and obtain the value of the integral

\[\int\limits_{-\pi}^\pi x^2\cdot\sin x\, \mathrm{d}x\,.\]

Solution

First we show that \(f(x) = -f(-x)\) holds:

\[f(-x) = (-x)^2\cdot\sin(-x)=x^2\cdot(-\sin(x)) = -f(x)\]

This can also be checked with Sage

and is confirmed by the shape of the graph of the function:

If the boundaries of an integral over an odd function are symmetric with respect to zero, as is the case here, the integral vanishes.

Sage can confirm that: