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 in the first quadrant.

Answer 1

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 $ABCD$.

Thus area of $ABCD$ $={\int }_1^{4}ydx$

$={\int }_1^4\sqrt{x}dx$

$={\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_1^4$

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

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

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