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.
$$\int_{-1}^1 |x^3-x| \; dx$$

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}
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