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