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Compute the area between the graphs of functions

by RoRi

Let

    \[ f(x) = x - x^2, \qquad g(x) = -x. \]

Find the area between the graphs of f and g on the interval [0,2].

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

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Then, we calculate the area, a(S) of S:

    \begin{align*} a(S) = \int_0^2 (f(x) - g(x))\, dx &= \int_0^2 (2x - x^2) \, dx \\ &= \left. 2 \frac{x^2}{2} \right|_0^2 - \left. \frac{x^3}{3} \right|_0^2 \\ &= 4 - \frac{8}{3} \\ &= \frac{4}{3}. \end{align*}

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