Let \(f\) and \(g\) be continuous on \([a,b],\) and suppose that \(f(x) \geq g(x)\) for all \(x\) in \([a,b].\) Then the area of the region between the graphs of \(f\) and \(g\) and the vertical lines \(x=a\) and \(x=b\) is
$$
A = \int_a^b [f(x) – g(x)] \; dx
$$
$$
A = \int_a^b [f(x) – g(x)] \; dx
$$
Evaluate the definite integral.
$$
\int_{-1}^1 |x^3-x| \; dx
$$
$$
\int_{-1}^1 |x^3-x| \; dx
$$
First note that
$$
\begin{aligned}
f(x) & = |x^3-x| \\
& = \begin{cases}
x^3-x, & \text{ if } -1 \leq x < 0 \\
-(x^3-x), & \text{ if } 0 \leq x \leq 1
\end{cases}
\end{aligned}
$$
So,
$$
\begin{aligned}
\int_{-1}^1 |x^3-x| \; dx &= \int_{-1}^0 x^3-x \; dx + \int_{0}^1 -(x^3-x) \; dx \\
& =\left( \frac{x^4}{4} - \frac{x^2}{2} \right) \Big|_{-1}^0 + \left(\frac{x^2}{2}- \frac{x^4}{4} \right) \Big|_0^1 \\
& = 1/2
\end{aligned}
$$