The area of the region bounded by the parabola $x^{2}=16 y, y=1, y=4$ and the Y-axis lying in the first…
The area of the region bounded by the parabola $x^{2}=16 y, y=1, y=4$ and the
Y-axis lying in the first quartrant is
$\frac{55}{3}$ sq. units
$\frac{56}{3}$ sq. units
$\frac{52}{3}$ sq. units
$\frac{53}{3}$ sq. units
Solution
The equation of curve is $x^{2}=16 y \Rightarrow x=4 \sqrt{y}$ Required area is shaded.
$\begin{aligned}
A &=\int_{1}^{4} 4 \sqrt{y} d y=4 \int_{1}^{4} y^{\frac{1}{2}} d y=4\left[\frac{y^{\frac{3}{2}}}{\left(\frac{3}{2}\right)}\right]_{1}^{4} \\
&=\frac{8}{3}(4 \sqrt{4}-1)=\frac{8 \times 7}{3}=\frac{56}{3}
\end{aligned}$
