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Compute the area between the graphs of functions

Let

    \[ f(x) = x^{1/2}, \qquad g(x) = x^2. \]

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.

Rendered by QuickLaTeX.com

For this exercise, since f and g cross in the interval we are interested in (and we are calculating the area between the graphs, so we’re really looking at the integral of |f(x) - g(x)|) we split the integral into two pieces (at the point the functions cross). We have f(x) \geq g(x) on [0,1] and g(x) \geq f(x) on [1,2]; thus, we have

    \begin{align*}   a(S) = \int_0^2 |f(x) - g(x)| \, dx &= \int_0^1 \left(x^{1/2} - x^2\right) \, dx + \int_1^2 \left(x^2 - x^{1/2}\right) \, dx \\   &= \left. \frac{2}{3}x^{3/2} \right|_0^1 - \left. \frac{x^3}{3} \right|_0^1 + \left. \frac{x^3}{3} \right|_1^2 - \left. \frac{2}{3}x^{3/2} \right|_1^2 \\   &= \left( \frac{2}{3} \right) - \left(\frac{1}{3} \right)+ \left( \frac{8}{3} - \frac{1}{3} \right) - \left( \frac{4 \sqrt{2}}{3} - \frac{2}{3} \right)  \\   &= \frac{10}{3} - \frac{4 \sqrt{2}}{3}. \end{align*}

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