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

Question

The area bounded by the parabola y2 = 4ax and the line x = a and x = 4a is (A) (35 a2/3) (B) (4 a2/3) (C) (7 a2/3) (D) (56 a2/3)

Tardigrade Admin · Accepted Answer

Required area = area of curve PSMNRQP =2 area of curve PSRQP =2 ∫ limitsa4 a √4 a x d x =4 √a[(x(3/2)/(3)2)]a4 a=(8/3) √a[(4 a) (3/2)-a(3/2)] =(8/3) √a[8 a(3/2)-a (3/2))=(56/3) a2

Q. The area bounded by the parabola $y^{2} = 4 a x$ and the line $x = a$ and $x = 4 a$ is

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

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

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

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

Required area = area of curve PSMNRQP
$= 2$ area of curve PSRQP
$= 2 a \int 4 a 4 a x d x$
$= 4 a [\frac{x ^{\frac{3}{2}}}{\frac{3}{2}}]_{a}^{4 a} = \frac{8}{3} a [(4 a)^{\frac{3}{2}} - a^{\frac{3}{2}}]$
$= \frac{8}{3} a [8 a^{\frac{3}{2}} - a \frac{3}{2}) = \frac{56}{3} a^{2}$

Q. The area bounded by the parabola y2=4ax and the line x=a and x=4a is

335a2​

34a2​

37a2​

356a2​