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Compute the area between the graphs of functions

by RoRi

Let

    \[ f(x) = x^{1/3}, \qquad g(x) = x^{1/2}. \]

Find the area between the graphs of f and g on the interval [0,1].

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_0^1 (f(x) - g(x))\, dx &= \int_0^1 \left(x^{1/3} - x^{1/2}\right) \, dx \\ &= \left. \frac{3}{4}x^{4/3} \right|_0^1 - \left. \frac{2}{3}x^{3/2} \right|_0^1 \\ &= \frac{3}{4} - \frac{2}{3}\\ &= \frac{1}{12}. \end{align*}

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