Discontinuities in intervals of integration

In summary, the conversation discusses different types of discontinuities and how to deal with them in integration, including infinite, jump, and removable discontinuities. The summation rule is still valid for removable discontinuities, and there may be special cases such as the Dirac delta. It is important to understand the context and purpose of the integration in order to determine the appropriate method. The conversation also mentions Lebesgue integration, which is typically taught in a real analysis course, and the difference between removable discontinuities and the Dirac delta. The conversation concludes by mentioning the complexity and variety of integration, and how advanced courses will provide a deeper understanding of the subject.
  • #1
PFuser1232
479
20
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?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
Simon Bridge said:
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?
 
  • #5
RUber said:
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?
 
  • #6
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.
 
  • Like
Likes Simon Bridge
  • #7
MohammedRady97 said:
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].
 
  • #8
RUber said:
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?
 
  • #9
MohammedRady97 said:
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
 
  • Like
Likes PFuser1232
  • #10
RUber said:
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.
 
  • Like
Likes RUber
  • #11
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.
 
  • Like
Likes PFuser1232

What are discontinuities in intervals of integration?

Discontinuities in intervals of integration refer to points in a function where there is a sudden jump or break in the graph. This can occur when there is a discontinuity in the function itself, such as a removable or non-removable discontinuity, or when there is a jump in the derivative of the function.

Why are discontinuities important in integration?

Discontinuities play a crucial role in integration because they can affect the convergence or divergence of the integral. If a function has a discontinuity in the interval of integration, the integral may not exist or may need to be evaluated using different techniques, such as using Cauchy's principal value.

How do you identify discontinuities in an interval of integration?

To identify discontinuities in an interval of integration, you can look for points where the function is undefined, where there is a jump in the graph, or where the derivative of the function is undefined. It is also important to check the endpoints of the interval for potential discontinuities.

How do discontinuities affect the value of the integral?

Discontinuities can affect the value of the integral in different ways. If the discontinuity is removable, meaning it can be filled in without changing the overall behavior of the function, the value of the integral will not be affected. However, if the discontinuity is non-removable, the integral may not exist or may need to be evaluated using special techniques.

Can discontinuities be positive or negative?

Yes, discontinuities can be positive or negative, depending on the behavior of the function at that point. For example, a jump discontinuity could have a positive or negative jump, and a removable discontinuity could have a positive or negative hole in the graph. It is important to consider the direction of the jump or hole when evaluating the integral.

Similar threads

I Integration with different infinitesimal intervals
    • Calculus
Replies
31
Views
941
I Substitution in a definite integral
    • Calculus
Replies
6
Views
1K
I Relating integration of forms to Riemann integration
    • Calculus
Replies
3
Views
2K
MHB How to prove this corollary in Line Integral using Riemann integral
    • Calculus
Replies
1
Views
1K
A Summing over continuum and uncountable numerocities
    • Calculus
Replies
1
Views
951
B Why is this definite integral a single number?
    • Calculus
Replies
20
Views
2K
Indefinite integral with discontinuous integrand
    • Calculus
Replies
6
Views
1K
A few questions to make sure I understand Fourier series
    • Calculus and Beyond Homework Help
Replies
3
Views
298
I Multivariable fundamental calculus theorem in Wald
    • Calculus
Replies
2
Views
1K
I Curiosity: there exists the exponential integral?
    • Calculus
Replies
3
Views
975

Hot Threads

Recent Insights

Back
Top