Compute the area between the graphs of functions

Let

$f(x) = |x|, \qquad g(x) = x^2-1.$

Find the area between the graphs of $f$ and $g$ on the interval $[-1,1]$ .

First, we draw the graph, shading the region $S$ between the two graphs in blue.

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In this problem $f(x) \geq g(x)$ on the entire interval $[-1,1]$ ; however, the function $f(x) = |x|$ is defined piecewise on this interval as

$f(x) = |x| = \begin{cases} -x & \text{if } x \in [-1,0) \\ x & \text{if } x \in [0,1] \end{cases}.$

Thus, we break the integral into intervals $[-1,0]$ and $[0,1]$ , and in both cases we are taking the integral of $f(x) - g(x)$ , but the definition of $f$ will depend on which interval we are in. The computation is as follows:

$\begin{align*} a(S) = \int_{-1}^{1} (f(x) - g(x)) \, dx &= \int_{-1}^0 (-x - x^2 + 1) \, dx + \int_0^1 (x - x^2 + 1) \, dx \\ &= \left. \left(-\frac{x^2}{2} - \frac{x^3}{3} + x \right) \right|_{-1}^0 + \left. \left( \frac{x^2}{2} - \frac{x^3}{3} + x \right) \right|_0^1 \\ &= \left( \frac{1}{2} - \frac{1}{3} + 1 \right) + \left( \frac{1}{2} - \frac{1}{3} + 1 \right) \\ &= \frac{7}{3}. \end{align*}$

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Compute the area between the graphs of functions

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