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 required area is bounded by the curve, the Y-axis, and the lines $y=2$ and $y=4$.

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

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

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

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

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

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