$\quad$ The area bounded by the parabola $x^{2}=4 y$ and the lines $y=2, \quad y=4$ and $Y$ -axis is
$\quad$ The area bounded by the parabola $x^{2}=4 y$ and the lines $y=2, \quad y=4$ and $Y$ -axis is
$\frac{4}{3}(8-2 \sqrt{2})$ sq. units
$\frac{8}{3}(8-2 \sqrt{2})$ sq. units
$\frac{8}{3}(8+2 \sqrt{2})$ sq. units
$(8-2 \sqrt{2})$ sq. units
Solution
Required area is shaded.
$\begin{aligned}
A &=2 \int_{2}^{4}(2 \sqrt{y}) d y \\
&=4 \int_{2}^{4} y^{\frac{1}{2}} d y=4\left[\frac{y^{\frac{3}{2}}}{\left(\frac{3}{2}\right)}\right]_{2}^{4}=\frac{8}{3}[4 \sqrt{4}-2 \sqrt{2}] \\
&=\frac{8}{3}(8-2 \sqrt{2})
\end{aligned}$
