Area between the curves Y=sinx and Y=[sinx]from x=0 to x=2pi is

let f(x) , g(x) be the two functions

f(x) = sinx

g(x) = [sinx]

area bw two curves is {f(x) - g(x)}dx  lim from 0  to pi

A = {sinx - [sinx]}dx        lim from 0 to pi

A = sinxdx       -     [sinx]dx                 lim 0 to pi

A =  I₁             -         I₂                             ..............................1

I₁ = sinxdx           lim 0 to pi

I₁ = -cosx         lim 0 to pi

I₁ = -[cospi-cos0] = 2 

I₂ = [sinx]dx            lim 0 to pi

    = [sinx]dx  lim 0 to pi/2 + [sinx]dx    lim pi/2 to pi

    = 0 + 0 = 0

A = I₁ - I₂ = 2units