Discontinuities in intervals of integration

[PFuser1232]

I want to make sure I understand how to deal with discontinuities when they lie within an interval of integration.

Infinite Discontinuities
In this case, do we always resort to improper integrals?

Jump Discontinuities
If there is a finite number of jump discontinuities in the integration interval, we could always use the following property:
$$\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$$

Removable Discontinuities
We deal with removable discontinuities in a similar way to how we deal with jump discontinuities. What I find confusing, though, is the fact that there are two types of removable discontinuities. The function may or may not be defined at a point of discontinuity. Does this make any difference? Or is the summation rule still valid in this case?

 

[RUber]

The summation rule is still valid. There may be special cases, like the Dirac delta, but most of the time, removing a point, or set of points (i.e. a set of measure zero), will not impact the total area under the curve.

 

[Simon Bridge]

In order to know what to do you have to know why you are doing the integral in the first place.
If the question is just "find the integral of..." then you have to infer from what "find the integral" means.
i.e. you need to understand what you are doing rather than just try to memorize the rules.

Remember that the idea of an integral is to find an area ... work out which area the problem needs you to find and you are there.

 

[PFuser1232]

In order to know what to do you have to know why you are doing the integral in the first place.
If the question is just "find the integral of..." then you have to infer from what "find the integral" means.
i.e. you need to understand what you are doing rather than just try to memorize the rules.

Remember that the idea of an integral is to find an area ... work out which area the problem needs you to find and you are there.

The only type of definite integral I'm familiar with is the Riemann integral. Is there another interpretation for the area under a curve?

 

[PFuser1232]

The summation rule is still valid. There may be special cases, like the Dirac delta, but most of the time, removing a point, or set of points (i.e. a set of measure zero), will not impact the total area under the curve.

There was a theorem in Stewart's Calculus about how we define integrability. "If $f$ is continuous on $[a,b]$, or if $f$ has a finite number of jump discontinuities in $[a,b]$, then $f$ is integrable on $[a,b]$."
Is this theorem incomplete? Isn't $f$ allowed to have a finite number of removable discontinuities as well?

 

[RUber]

I think Simon Bridge was right in that it depends on the application. Take a look at Lebesgue integration if you want to be free to remove discontinuities indiscriminately. The jump discontinuities ensure that the function is still finite over the interval, where a removable discontinuity has more possibilities.
Also, look at the Dirac delta for another example of why just saying "removable discontinuity" is not sufficient.

 

[JonnyG]

There was a theorem in Stewart's Calculus about how we define integrability. "If ff is continuous on [a,b][a,b], or if ff has a finite number of jump discontinuities in [a,b][a,b], then ff is integrable on [a,b][a,b]."
Is this theorem incomplete? Isn't ff allowed to have a finite number of removable discontinuities as well?

The function f can have up to a countable number of discontinuities as long as it is bounded over [a,b].

 

[PFuser1232]

I think Simon Bridge was right in that it depends on the application. Take a look at Lebesgue integration if you want to be free to remove discontinuities indiscriminately. The jump discontinuities ensure that the function is still finite over the interval, where a removable discontinuity has more possibilities.
Also, look at the Dirac delta for another example of why just saying "removable discontinuity" is not sufficient.

Will I learn about Lebesgue integration in Calculus II? Or will I have to wait till analysis?

 

[JonnyG]

Will I learn about Lebesgue integration in Calculus II? Or will I have to wait till analysis?

You will have to wait to take a real analysis course. And not a Spivak level course, but a course that would, for example, use Rudin or Pugh

 

[micromass]

Also, look at the Dirac delta for another example of why just saying "removable discontinuity" is not sufficient.

The Dirac delta is not really a function in the classical sense. In either case, it would be unfair to say that the Dirac delta has a removable discontinuity. That is not what is meant with the term removable discontinuity.

 

[Simon Bridge]

The only type of definite integral I'm familiar with is the Riemann integral. Is there another interpretation for the area under a curve?

That is OK for now - but there are lots of different kinds of curves. What you have to do for the integration will depend on the curve type and what the context is.
i.e. a curve may have large gaps where the function is undefined - in general you cannot integrate this; but the problem may only be interested in the area where the curve is defined or it may be possible to do the integration in a coordinate system where the gaps vanish or are unimportant.

You'll notice that there are very big books written about calculus - even just those about how to do integrations.
There is no way to give a totally inclusive and complete answer, in detail, which also fits into the box.
You will learn more in the advanced courses.