Integral of a floor function?

[rachaelmw]

I was doing well with my homework assignment until I saw this:

1. Define:
g(x) = [
x] ? 2[
x/2], where [x] denotes the floor function

2. Evaluate the integral:
Int g(x)

While I understand the concept of the floor function, I have absolutely no idea how to define or integrate it. I have looked in my old calc textbooks, but to no avail. Any help is appreciated.

 

[pka]

Is this the problem?
\(\displaystyle \begin{array}{l} g(x) = \left\lfloor x \right\rfloor - 2\left\lfloor {\frac{x}{2}} \right\rfloor \\ \int\limits_a^b {g(x)dx} \\ \end{array}\)
If so you did not give value of the limits a&b.

Otherwise, g(x) has no general antiderivative.

 

[rachaelmw]

Yes, you are correct. It is a definite integral from 0 to 2.

 

[pka]

From the image you see how the function works.
The integral will be the total area under the graph between 0 and 2.