Question

The area bounded by the parabola $x^2=4ay,x$-axis and the straight line $y=2a$ is:

• A. $16\sqrt{2}{a}^2$ sq. units
• B. $\frac{16\sqrt{2}}{3}{a}^2$ sq. units
• C. $\frac{32\sqrt{2}}{3}{a}^2$ sq. units
• D. $\frac{32\sqrt{2}}{5}{a}^2$ sq. units

Answer 1

The correct option is B $\frac{16\sqrt{2}}{3}{a}^2$ sq. units

So, area of the shaded region
$-\stackrel{2\sqrt{2}a}{\underset{0}{\int }}{x}^2/4adx+\stackrel{2\sqrt{2}a}{\underset{0}{\int }}\left(2a\right)dx$
$2a\left(2\sqrt{2a}\right)-\frac{}{4a}\frac{x^3}{3}{|}_0^{2\sqrt{2a}}$
$=4\sqrt{2}{a}^2-\frac{(2^{3/2}a{)}^3}{4a\times 3}$
$4\sqrt{2}{a}^2-\frac{2\times 2\sqrt{2}{a}^2}{3}$
$=\frac{16\sqrt{2}{a}^2}{3}squnits.$