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 ![[1,2]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-7268b2f31e88f380a9097cbbcb097a29_l3.png "Rendered by QuickLaTeX.com")
.
First, we draw the graph, shading the region 
between the two graphs in blue.
Then, we calculate the area, 
of :
![\begin{align*} a(S) = \int_1^2 (g(x) - f(x))\, dx &= \int_1^2 \left(x^{1/2} - x^{1/3}\right) \, dx \\ &= \left. \frac{2}{3}x^{3/2} \right|_1^2 - \left. \frac{3}{4}x^{4/3} \right|_1^2 \\ &= \left(\frac{4 \sqrt{2}}{3} - \frac{2}{3} \right) - \left( \frac{6 \sqrt[3]{2}}{4} - \frac{3}{4} \right) \\ &= \frac{4 \sqrt{2}}{3} - \frac{3 \sqrt[3]{2}}{2} + \frac{1}{12}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-0229a5abd5e31d8a5bdca7e534b70ab8_l3.png "Rendered by QuickLaTeX.com")
Let
Find the area between the graphs of and on the interval .
First, we draw the graph, shading the region between the two graphs in blue.
Then, we calculate the area, of :