$\frac {1+sgn(sinx)}{2}$

Find the area bounded by the function $y=\frac {1+sgn(sinx)}{2}$ with x-axis from 0 to $2\pi$ when sgn represents signum or sign function.

Solution

For $0 < x < \pi$, sinx > 0, so sgn(sinx)=1 giving y=1.

For $\pi < x < 2\pi$, sinx < 0, so sgn(sinx)=-1 giving y=0.

So, y is a square wave.

Area bounded with x-axis = area of rectangle + 0 = $\pi.1 + 0 = \pi$ sq. unit