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Question

Mathematics Question on Area between Two Curves

The area bounded by the parabola y2=4axy^2 = 4ax and the line x=ax = a and x=4ax = 4a is

A

35a23\frac{35 a^2}{3}

B

4a23\frac{4 a^2}{3}

C

7a23\frac{7 a^2}{3}

D

56a23\frac{56 a^2}{3}

Answer

56a23\frac{56 a^2}{3}

Explanation

Solution

Required area = area of curve PSMNRQP
=2=2 area of curve PSRQP
=2∫a4a4axdx=2 \int\limits_{a}^{4 a} \sqrt{4 a x} d x
=4a[x3232]a4a=83a[(4a)32−a32]=4 \sqrt{a}\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{a}^{4 a}=\frac{8}{3} \sqrt{a}\left[(4 a)^ \frac{3}{2}-a^{\frac{3}{2}}\right]
=83a[8a32−a32)=563a2=\frac{8}{3} \sqrt{a}\left[8 a^{\frac{3}{2}}-a \frac{3}{2}\right)=\frac{56}{3} a^{2}