Question

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

Answer 1

The area of the region bounded by the curve $x^2=4y,y=2$ and $y=4$, and the $y$-axis is the area $ABCD$.
Area of $ABCD$ $={\int }_2^{4}xdy$
$={\int }_2^{4}2\sqrt{y}dy$
$=2{\int }_2^4\sqrt{y}dy$
$=2{\left[\frac{y^{\frac{3}{2}}}{\frac{3}{2}}\right]}_2^4$
$=\frac{4}{3}\left[\right(4{)}^{\frac{3}{2}}-\left(2{)}^{\frac{3}{2}}\right]$
$=\frac{4}{3}[8-2\sqrt{2}]$
$=\left(\frac{32-8\sqrt{2}}{3}\right)units$