Question

Find the area of the region bounded by $y^2=9x,x=2,x=4$ and the $x$-axis in the first quadrant.

Answer 1

The area of the region bounded by the curve, $y^2=9x,x=2$ and $x=4$, and the $x$-axis is the area $ABCD$.
Area of $ABCD$ $={\int }_2^{4}ydx$
$={\int }_2^{4}3\sqrt{x}dx$
$=3{\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_2^4$
$=2[x^{\frac{3}{2}}{]}_2^4$
$=2\left[\right(4{)}^{\frac{3}{2}}-\left(2{)}^{\frac{3}{2}}\right]$
$=2[8-2\sqrt{2}]$
$=(16-4\sqrt{2})units$