The area bounded by the curve `a^2y=x^2(x+a)` and the x-axis is `(a^2)/3s qdotu n i t s` (b) `(a^2)/4s qdotu n i t s` `(3a^2)/4s qdotu n i t s` (d) `(a^2)/(12)s qdotu n i t s`

Options

A. `a^(2)//3` sq. units
B. `a^(2)//4` sq. units
C. `3a^(2)//4` sq. units
D. `a^(2)//12` sq. units

Answer

D

Explanation

The curve is `y=(x^(2)(x+a))/(a^(2))`, which is a cubic polynomial.
Since `(x^(2)(x+a))/(a^(2))=0` has a repeated root x=0, it touchses x-axis at (0,0) and intersects at (-a,0).

`"Required area "=int_(-a)^(0)ydx `
`=int_(-a)^(0)[(x^(2)(x+a))/(a^(2))]dx`
`=a^(2)//12` sq. units.