Answer
$\displaystyle \frac{2}{5}(32768)-\frac{6}{11}(2048)≈11990.10909$
Work Step by Step
$\displaystyle \int_0^{64}\sqrt{u}(u-\sqrt[3] u)du=\int_0^{64}u^{1/2}(u-u^{1/3})du=\int_0^{64}u^{3/2}-u^{5/6}du$
$\displaystyle (\frac{u^{5/2}}{5/2}-\frac{u^{11/6}}{11/6})|_0^{64}$
$\displaystyle (\frac{2}{5}u^{5/2}-\frac{6}{11}u^{11/6})|_0^{64}$
$\displaystyle [\frac{2}{5}(64)^{5/2}-\frac{6}{11}(64)^{11/6}]-[\frac{2}{5}(0)^{5/2}-\frac{6}{11}(0)^{11/6}]$
$\displaystyle [\frac{2}{5}(8)^{5}-\frac{6}{11}(2)^{11}]-[0]$
$\displaystyle \frac{2}{5}(32768)-\frac{6}{11}(2048)≈11990.10909$
