Q.
The figure shows a $\mathrm{\Delta}AOB$ and the parabola $y = x^{2}.$ The ratio of the area of the $\mathrm{\Delta}AOB$ to the area of the region $AOB$ of the parabola $y = x^{2}$ is equal to
A) $\frac{3}{5}$
B) $\frac{3}{4}$
C) $\frac{7}{8}$
D) $\frac{5}{6}$
Explanation
(b)
$Area\ of\ curve\ OAB = 2\int_{0}^{a^{2}}{x\ dy}$
[image:8_image59.jpeg]
$= 2\int_{0}^{a^{2}}\sqrt{y}dy = 2\left\lbrack \frac{y^{3/2}}{3/2} \right\rbrack_{0}^{a^{2}}$
$= \frac{4}{3}\left\lbrack a^{3} \right\rbrack$
$Now,\ Area\ of\ \mathrm{\Delta}OAB = \frac{1}{2} \times AB \times OC$
$= \frac{1}{2} \times 2a \times a^{2} = a^{3}$
$\therefore\frac{Area\ of\ \mathrm{\Delta}\ AOB}{Area\ of\ curve\ AOB} = \frac{a^{3}}{\frac{4}{3}a^{3}} = \frac{3}{4}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.