# The area under a discontinuous & integrable function

1. Mar 12, 2012

### shtephy

Does it make sense to talk about the area of the region {(x,y)|x$\in$[a,b];y$\in$[0,f(x)]} for a positive function f defined on an interval [a,b], where a,b$\in$ℝ and f is integrable on that interval, even if the function is not continuous?

Last edited: Mar 12, 2012
2. Mar 12, 2012

### A. Bahat

It depends on what you mean by area. It is common to define this area as, in fact, the integral of f from a to b. In that case it makes perfect sense to speak of the area under an integrable, discontinuous function.

3. Mar 12, 2012

### shtephy

Do you all agree with this?

My teacher says that the notion of area can be used only for bounded regions and we can say that the integral form a to b is that area only if the function is continuous, because the surface is limited by the lines x=a, x=b, Ox and the graph of f and, if f is discontinuous, then its graph will not wrap the upper part of the region completely.

http://img207.imageshack.us/img207/5329/imagu.png [Broken]

She also says that open disks don’t have area.

On the other hand, I think that the existence of area depends only on the set of points, and, following the definition on Wikipedia (go to Formal Definition, the last point), the set {(x,y)|x$\in$[a,b];y$\in$[0,f(x)]} should have area.

Last edited by a moderator: May 5, 2017
4. Mar 12, 2012

### Char. Limit

If the set of discontinuous points in an interval is finite, then the function can be integrated over that interval, I believe.

Last edited: Mar 12, 2012
5. Mar 12, 2012

### shtephy

It is, because the function is integrable.
It can, for the same reason.

The problem is whether we can talk about the area of that surface or not.

6. Mar 12, 2012

### micromass

Staff Emeritus
Yes, we talk about area there. The area is just defined as the integral $\int_a^b |f(x)|dx$. There is no reason not to call that the area.

And open disks DO have an area. I don't get where your teacher is getting all that nonsense.

7. Mar 12, 2012

### shtephy

Thank you.

8. Mar 12, 2012

### mathwonk

well maybe your teacher is defining area in only a restricted sense, to make your life simpler. ask him/her.