Find the area of the region bounded by the parabola `y^2=4ax`, its axis and two ordinates `x=4` and `x=9`

Options

A. `(86sqrt(b))/(3)` sq units
B. `(76sqrt(a))/(3)` sq units
C. `(92sqrt(a))/(4)` sq units
D. `(93sqrt(b))/(3)` sq units

Answer

B

Explanation

Given equation of parabola is `y^(2) = 4ax`, its axis is Y = 0 and vertex is (0, 0). Also, given ordinates x = 4 and x = 9. The bounded region in I quadrant is ABCDA.
`therefore` Required area
`=int_(4)^(9)ydx=int_(4)^(9)sqrt(4ax)dx " "[becausey^(2)=4aximpliesy=sqrt(4ax),"as y is in I quadranr]"`
`=2sqrt(a)int_(4)^(9)sqrt(x)dx=2sqrt(a)[(x^(3//2))/(3//2)]_(4)^(9)`
`=2sqrt(a)xx(2)/(3)[(9)^(3//2)-(4)^(3//2)]`
`=(4)/(3)sqrt(a)[(3^(2))^(3//2)-(2^(2))^(3//2)]`
`=(4)/(3)sqrt(a)[27-8]=(4)/(3)sqrt(a)(19)=(76)/(3)sqrt(a)` sq unit
Hence, the required area is `(76sqrt(a))/(3)` sq unit.