7.1 | Area of a Region Between Two Curves


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
$$
A = \int_a^b [f(x) – g(x)] \; dx
$$
Evaluate the definite integral.
$$
\int_{-1}^1 |x^3-x| \; dx
$$


OLAOFARBTC71E1

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, $$ \begin{aligned} \int_{-1}^1 |x^3-x| \; dx &= \int_{-1}^0 x^3-x \; dx + \int_{0}^1 -(x^3-x) \; dx \\ & =\left( \frac{x^4}{4} - \frac{x^2}{2} \right) \Big|_{-1}^0 + \left(\frac{x^2}{2}- \frac{x^4}{4} \right) \Big|_0^1 \\ & = 1/2 \end{aligned} $$

E 7.1 Exercises