Find the area of the region bounded by the curve
Question:

Find the area of the region bounded by the curve $y^{2}=x$ and the lines $x=1, x=4$ and the $x$-axis.

Solution:

The area of the region bounded by the curve, $y^{2}=x$, the lines, $x=1$ and $x=4$, and the $x$-axis is the area $A B C D$.

Area of $\mathrm{ABCD}=\int_{1}^{4} y d x$

$=\int_{1}^{4} \sqrt{x} d x$

$=\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{1}^{4}$

$=\frac{2}{3}\left[(4)^{\frac{3}{2}}-(1)^{\frac{3}{2}}\right]$

$=\frac{2}{3}[8-1]$

$=\frac{14}{3}$ units