Finding an area under the curve with no area

Question

Hello I am trying to explain why the function h(x)

$h(x)= x^{99} + x$

will always have the area without determining an antiderivative of h(x) that we must have

First I thought of drawing the graph and noticed that the area under the curve on the right side will be identical to the area above the curve on the left side.

I am not sure how to explain in detail why the area bounded by b,-b will always equal to 0.

Answer 1

Clearly $h(x)$ is an odd function, that is $h(x)=-h(-x)$ for all $x$. Thus we have $$\int_{-b}^{b}h(x)dx=\int_{0}^{b}h(x)dx+\int_{-b}^{0}h(x)dx$$ $$=\int_{0}^{b}h(x)dx-\int_{-b}^{0}h(-x)dx$$ $$=\int_{0}^{b}h(x)dx+\int_{b}^{0}h(x)dx$$ $$=\int_{0}^{b}h(x)dx-\int_{0}^{b}h(x)dx=0.$$