Mathematics - Area Under Curves Question with Solution | TestHub

The given curve is $y=4x^2$. In the first quadrant, $x=\frac{\sqrt{y}}{2}$.

The area of the region bounded by the curve, the y-axis ($x=0$), and the lines $y=2$ and $y=4$ is given by:

$A={\int }_2^{4}xdy$

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

$A=\frac{1}{2}{\left[\frac{y^{3/2}}{3/2}\right]}_2^4$

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

$A=\frac{1}{3}[8-2\sqrt{2}]$ sq. units.

Answer: $\frac{1}{3}[8-2\sqrt{2}]$ sq.units