Answer
$Area = \frac{64}{5}$
Work Step by Step
$y = x\sqrt{x}$
Under the graph on the interval $0 \leq x \leq 4$, we can see that the area is less than the area of a triangle with base 4 and height 8. The area of this triangle is $\frac{1}{2}(4)(8) = 16$. Thus the area under the graph is somewhat less than 16 square units.
We can find the exact area $A$ under the graph:
$A = \int_{0}^{4}x\sqrt{x}~dx$
$A = \int_{0}^{4}x^{3/2}~dx$
$A = \frac{2}{5}x^{5/2}~\Big\vert_{0}^{4}$
$A = \frac{2}{5}(4)^{5/2}- \frac{2}{5}(0)^{5/2}$
$A = \frac{2}{5}(32)- 0$
$A = \frac{64}{5}$
