What is the integral equal to? #2

Question

Suppose f(x) is the greatest integer function, i.e., f(x) equals the greatest integer less than or equal to x.
So for example f(2.3)=2, f(4)=4, and f(6.9)=6. Find

 
∫^7  f(x)dx.

 3

 

What is the integral equal to? 

Answer 1

The integral is from x =3 to x = 7

Split into the intervals [3,4), [4,5),[5,6),[6,7)

In each of these intervals, the width is 1, and the heights of the boxes are 3,4,5,6 respectively.

The area = 1 x 3 + 1 x 4 + 1x 5 + 1 x 6 = 18

Answer 2

This integral can be done by hand, if it is an integer formula then that means from 1 to 1.9999... all the values will be 1, and the dx of this is 1, so the way to integrate this is to have 1*1+1*2+1*3+1*4+1*5+1*6+1*7 and then subtract the lower boundary of 1*1+1*2+1*3 and you will have 7+6+5+4 = 22. So the ∫⌊x⌋dx [3,7] = 22. Also ⌊x⌋ is known as Floor Function or the greatest integer function.