Find the area bounded by the curve `y=x(x-1)(x-2)` and the x-axis.

Explanation

The given curve crosses the X-axis at points (0, 0), (1, 0) and (2, 0).
Its graph is shown in the figure.
Area of the region bounded by curve and X-axis
`=` Area of OACO + Area of ADBA
`=int_(0)^(1)"y "dx+int_(1)^(2)"(-y) "dx`
`=int_(0)^(1)4x(x-1)(x-2)dx-int_(1)^(2)4x(x-1)(x-2)dx`
`=4int_(0)^(1)(x^(3)-3x^(2)+2x)dx-4int_(1)^(2)(x^(3)-3x^(2)+2x)dx`
`=[((1)/(4)-1+1)-(0-0+0)]-4[(4-8+4)-((1)/(4)-1+1)]`
`=1+1=2 sq.` units.