Definite Integration Question 121
Question: Area inside the parabola $ y^{2}=4ax, $ between the lines $ x=a $ and $ x=4a $ is equal to
[Pb. CET 2002; Karnataka CET 2005]
Options:
A) $ 4a^{2} $
B) $ 8a^{2} $
C) $ 28\frac{a^{2}}{3} $
D) $ 35\frac{a^{2}}{3} $
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Answer:
Correct Answer: C
Solution:
We have $ y^{2}=4ax $
therefore $ y=2\sqrt{ax} $
We know the equations of lines $ x=a $ and $ x=4a $
The area inside the parabola between the lines $ A=\int_a^{4a}{ydx}=\int_a^{4a}{2\sqrt{ax}}dx=2\sqrt{a}\int_a^{4a}{{x^{\frac{1}{2}}}dx=2\sqrt{a}[ \frac{{x^{\frac{3}{2}}}}{\frac{3}{2}} ]}_a^{4a} $
$ =\frac{4}{3}{a^{\frac{1}{2}}}[ {{(4a)}^{\frac{3}{2}}}-{{(a)}^{\frac{3}{2}}} ]=\frac{4}{3}{a^{\frac{1}{2}}}{a^{\frac{3}{2}}}[8-1] $
$ =\frac{28}{3}a^{2} $ .
BETA
