The area bounded by $$f\left( x \right) = {x^2},\,0 \leqslant x \leqslant 1,\,g\left( x \right) = - x + 2,\,1 \leqslant x \leqslant 2$$          and $$x$$-axis is :

Solution :
Required area $$=$$ Area of $$OAB\,+$$   Area of $$ABC$$

Now, Area of $$OAB$$
$$\eqalign{ & = \int\limits_0^1 {f\left( x \right)dx} + \int\limits_1^2 {g\left( x \right)dx} \cr & = \int\limits_0^1 {{x^2}dx} + \int\limits_1^2 {\left( { - x + 2} \right)dx} \cr & = \left. {\frac{{{x^3}}}{3}} \right|_0^1 + \left[ {\frac{{ - {x^2}}}{2} + 2x} \right]_1^2 \cr & = \frac{1}{3} + \left[ {\left( {\frac{{ - 4}}{2} + 4} \right) - \left( {\frac{{ - 1}}{2} + 2} \right)} \right] \cr & = \frac{1}{3} + \frac{1}{2} \cr & = \frac{5}{6}{\text{ sq}}{\text{. unit}} \cr} $$