Sol. The curve y = 2x –x 2 represents a parabola opening downward and crossing x- axis at (0, 0) and (2, 0). Clearly, y= –x represents a line passing through the origin and making 135º with x- axis. A rough sketch of the two curves is shown in fig. The region whose area is to be found is shaded in fig. Here, we slice the shaded region into vertical strips. For the approximating rectangle shown in fig., we have

Width = Δ x Length = (y 1 –y 2 )

∴ Area = (y 1 –y 2 ) Δ x

The approximating rectangle can move horizontal between x = 0 and x= 3.

∴ required area =

=

=

= = – = sq. units.