A) \[4{{a}^{2}}\]
B) \[8{{a}^{2}}\]
C) \[28\frac{{{a}^{2}}}{3}\]
D) \[35\frac{{{a}^{2}}}{3}\]
Correct Answer: C
Solution :
We have \[{{y}^{2}}=4ax\] Þ \[y=2\sqrt{ax}\] We know the equations of lines \[x=a\] and \[x=4a\] \ The area inside the parabola between the lines \[A=\int_{a}^{4a}{y\,dx}=\int_{a}^{4a}{2\sqrt{ax}}\,dx=2\sqrt{a}\int_{a}^{4a}{\,{{x}^{\frac{1}{2}}}dx=2\sqrt{a}\left[ \frac{{{x}^{\frac{3}{2}}}}{\frac{3}{2}} \right]}_{a}^{4a}\] \[=\frac{4}{3}{{a}^{\frac{1}{2}}}\left[ {{(4a)}^{\frac{3}{2}}}-{{(a)}^{\frac{3}{2}}} \right]=\frac{4}{3}{{a}^{\frac{1}{2}}}{{a}^{\frac{3}{2}}}[8-1]\] \[=\frac{28}{3}{{a}^{2}}\].
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