Question

Find the area of the region lying in the first quadrant and bounded by
$y=4x^2,x=0,y=1$ and $y=4$.

Answer 1

The area in the first quadrant bounded by $y=4x^2,x=0,y=1$, and $y=4$ is represented by the shaded area $ABCDA$

$\therefore AreaABCD={\int }_1^{4}xdx$

$={\int }_1^4\frac{\sqrt{y}}{2}dx$

$=\frac{1}{2}{\left[\frac{y^{\frac{3}{2}}}{\frac{3}{2}}\right]}_1^4$

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

$=\frac{1}{3}[8-1]$

$=\frac{7}{3}$ sq. units