In general, the area enclosed between a curve and the x-axis is given by

A=\int_{a}^{b}\left|f\left(x\right)\right|\differentialD x\quad📖

since any region below the x-axis has f\left(x\right)<0, but area must always be positive.

This can be done with technology, or by splitting the integral into parts - where f is positive and where f is negative:

A=\int_{a}^{m}f\left(x\right)\differentialD x+\int_{m}^{b}-f\left(x\right)\differentialD x\quad🚫

Example

Find the area, shaded in the diagram below, enclosed between the curve of y=x^3-x and the x-axis.

We see from the graph that y is positive for -1<x<0 and negative for 0<x<1:

\begin{align}A & =\int_{-1}^1\left|x^3-x\right|\differentialD x\\ \placeholder{} & \placeholder{}\\ \placeholder{} & =\int_{-1}^0\left(x^3-x\right)\differentialD x+\int_0^1-\left(x^3-x\right)\differentialD x\\ \placeholder{} & \placeholder{}\\ \placeholder{} & =\left\lbrack\frac{x^4}{4}-\frac{x^2}{2}\right\rbrack_{-1}^0-\left\lbrack\frac{x^4}{4}-\frac{x^2}{2}\right\rbrack_0^1\\ \placeholder{} & \placeholder{}\\ \placeholder{} & =\left\lbrack0-\left(\frac14-\frac12\right)\right\rbrack-\left\lbrack\left(\frac14-\frac12\right)-0\right\rbrack\\ \placeholder{} & \placeholder{}\\ \placeholder{} & =\boxed{\frac12}\end{align}

Integration