Question

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

Answer 1

Let $AB$ represent line $y=2$ and $CD$ represent line $y=4$

and $CBOAD$ represent the curve $x^2=4y$

We have to find the area of $BCFE$

Area of $BCFE={\int }_2^{4}x.dy$

We know that $x^2=4y$

$\Rightarrow x=\pm \sqrt{4y}=\pm 2\sqrt{y}$

Since $BCFE$ is in first quadrant

$\therefore x=2\sqrt{y}$

Area of $BCFE={\int }_2^{4}x.dy$

$={\int }_2^{4}2\sqrt{y}.dy$

$=2{\int }_2^4\sqrt{y}.dy$

$=2{\left[\frac{y^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]}_2^4$

$=2{\left[\frac{y^{\frac{3}{2}}}{\frac{3}{2}}\right]}_2^4$

$=2\times \frac{2}{3}[4^{\frac{3}{2}}-2^{\frac{3}{2}}]$

$=\frac{4}{3}[8-2\sqrt{2}]$

Thus, area$=\frac{32-8\sqrt{2}}{3}$sq.units.