# Finding an area under the curve with no area

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.

• Perhaps the magic word is antisymmetric Oct 2, 2020 at 10:12
• Also, it's a polynomial so integration should be a no-brainer, right? Oct 2, 2020 at 10:15
• It is an odd function, that's why the integral is zero Oct 2, 2020 at 10:17

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