Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What is ₀^2 h(x) d x equal to?

Step-by-step solution

array ll Given, & f(x)=|x-1| ...(i) & g(x)=[x] ...(ii) and & h(x)=f(x) g(x) & h(x)=h(x)-1 [x] ...(iii) array aligned ₀^2 h(x) d x & = ₀^1|x-1|[x] d x+ ₁^2|x-1|[x] d x & = ₀^1(1-x) (0) d x+ ₁^2(x-1) (1) d x & = ₀^2(x-1) d x= [ x^2 2 -x ]₁^2 & = ( 4 2 -2 )- ( 1 2 -1 )= 1 2 aligned