What is the area under the curve $$y = \left| x \right| + \left| {x - 1} \right|$$    between $$x = 0$$  and $$x = 1\,?$$

Solution :
$$\eqalign{ & \left| x \right|\,{\text{for }}x \geqslant 0 \cr & = x{\text{ and }}\left| {x - 1} \right|{\text{ for }}x \leqslant 1 = - \left( {x - 1} \right), \cr & {\text{So, }}\int_0^1 {\left( {\left| x \right| + \left| {x - 11} \right|} \right)} = {\text{required area}} \cr & a = \int_0^1 {x\,dx} - \int_0^1 {\left( {x - 1} \right)dx} \cr & \,\,\,\,\, = \left[ {\frac{{{x^2}}}{2}} \right]_0^1 - \left[ {\frac{{{x^2}}}{2} - x} \right]_0^1 \cr & \,\,\,\,\, = \frac{1}{2} - \left( {\frac{1}{2} - 1} \right) \cr & \,\,\,\,\, = 1{\text{ sq}}{\text{. unit}} \cr} $$