Integrate Least Integer Function

Integrate Least Integer Function

What is the essence of the topic?

         To integrate a function is to find the area between the function and the axis(whichever we choose as per our convenience, either x or y). So let us have a look at the graph of least integer function which will give us a bit of idea of what we have to do?

Graph of Least Integer Function using Desmos

     The graph shows the function's behavior. This function return the least integer greater than the number which is input.

There are two interesting aspects to this graph:

1. The graph is discontinuous at integers.
2. The area under graph between two integers is basically finding the area of the rectangle. How? Have a look. The graph from 0 to 1 is a line y =1 except at x =0. Consider the x axis as one side of the rectangle, the function being the other line, x = 0, x = 1 are the imaginary sides. Now integrating from 0 to 1 is basically finding area of this rectangle.

   

   An illustration of the above point.

   Similarly for 1 to 2 y =2, x-axis, x=1,x=2 form the four sides.

Now let's integrate.

       We begin by taking the function and proceed as given below.

This is how you should proceed.

This is how you integrate the least integer function.