Question

The figure shows as triangle AOB and the parabola $y=x^2$. The ratio of the area of the triangle 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. $\frac{5}{6}$

Answer 1

Answer: (B)

Area of $\Delta AOB=\frac{1}{2}\times$ base $\times$ height
$=\frac{1}{2}\times 2a\times a^2=a^3$ units
Area of region AOB

$=2\underset{0}{\overset{a^2}{\int }}xdy=\underset{0}{\overset{a^2}{\int }}\sqrt{y}dy$
$=2{\left[\frac{y^{3/2}}{3/2}\right]}_0^{a^2}=\frac{4}{3}{a}^3$ units
$\therefore$ ratio of areas $=\frac{a^3}{\frac{4}{3}{a}^3}=\frac{3}{4}$