Let
![\[ f(x) = x^{1/3}, \qquad g(x) = x^{1/2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7a848a7e2d72998404908e92638271f0_l3.png "Rendered by QuickLaTeX.com")
Find the area between the graphs of 
and 
on the interval ![[0,2]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-d2292f5d2ce11b1d657528e0cf2e91a6_l3.png "Rendered by QuickLaTeX.com")
.
First, we draw the graph, shading the region 
between the two graphs in blue.
For this exercise, since and cross in the interval we are interested (and we are calculating the area between the graphs, so we’re really looking at the integral of 
) we split the integral into two pieces (at the point the functions cross).
![\[ a(S) = \int_0^2 |f(x) - g(x)| \, dx &= \int_0^1 \left(x^{1/3} - x^{1/2} \right) \, dx + \int_1^2 \left(x^{1/2} - x^{1/3}\right) \, dx \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ccc3d59c3c2ca58c6b1e53444ac108d0_l3.png "Rendered by QuickLaTeX.com")
We’ve already computed these integral in the previous two exercises (here and here). Using the results of those exercises we then have,
![\begin{align*} \int_0^1 \left( x^{1/3} - x^{1/2} \right) \, dx + \int_1^2 \left(x^{1/2} - x^{1/3} \right) \, dx &= \frac{1}{12} + \frac{4 \sqrt{2}}{3} - \frac{3 \sqrt[3]{2}}{2} + \frac{1}{12} \\ &= \frac{4 \sqrt{2}}{3} - \frac{3 \sqrt[3]{2}}{2} + \frac{1}{6}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-61297ea42dfda475fe24b281cb0a879c_l3.png "Rendered by QuickLaTeX.com")