question_answer

The area bounded by the parabola \[{{y}^{2}}=4\text{ }ax\] and the line \[x=a\] and \[x=4\,a\] is :

A)  \[\frac{35\,{{a}^{2}}}{3}\]                           

B)  \[\frac{4\,{{a}^{2}}}{3}\]

C)  \[\frac{7\,{{a}^{2}}}{3}\]                             

D)  \[\frac{56\,{{a}^{2}}}{3}\]

Correct Answer: D

Solution :

Required area                 = Area of curve PSMNRQP = 2 Area of curve PSRQP \[=2\int_{a}^{4a}{\sqrt{4ax}\,dx}\] \[=4\sqrt{a}\left[ \frac{{{x}^{3/2}}}{3/2} \right]_{a}^{4a}=\frac{8}{3}\sqrt{a}\left[ {{(4a)}^{3/2}}-{{a}^{3/2}} \right]\] \[=\frac{8}{3}\sqrt{a}\left( 8{{a}^{3/2}}-{{a}^{3/2}} \right)=\frac{56}{3}{{a}^{2}}\]