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Compute the area between the graphs of functions

Let

    \[ f(x) = 4-x^2, \qquad g(x) = 0. \]

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


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_{-2}^2 (f(x) - g(x)) \, dx &= \int_{-2}^2 (4-x^2) \, \\   &= \left.4x \biggr\rvert_{-2}^2 - \left. \frac{x^3}{3} \right|_{-2}^2 \\   &= (8+8) - \left( \frac{8}{3} + \frac{8}{3} \right) \\   &= \frac{32}{3}. \end{align*}

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