Thread: Derivative of a definite integral

Hi, colleagues!!

Where in my Calculus book do i have to search in order to find an exhaustive explanation of how derivative of a definite integral can be calculated?

Thank you all!

PS: Forgive me for my fool question!

We are working on this in my calc class too. Look at Riemann sums. That gives you the formula and info that you need.

The value of a definite integral is just a number, hence "definite" integral. So the derivative is 0. But I'm guessing that's not what you mean. In general the derivative of an integral is just the integrand since the derivative undoes the integral. So more specifically, what are you after? What situation did this come up in?

Thank you, man!!

..because I can't understand the issue when integral extremes depend on the derivative variable...

do you mean something like... , where one of the limits on the integral is the variable found in the integrand?

No problem cat glad I could help. Chemboy is right about that formula and that is what I was talking about.

Originally Posted by Chemboy

do you mean something like... , where one of the limits on the integral is the variable found in the integrand?

..something like d/dx of , where a = a(x,y) and b= b(x,y)

There you go cats that is correct! You understand integrals!! Congrats man.

Originally Posted by the man of science

There you go cats that is correct! You understand integrals!! Congrats man.

Sorry, I haven't understood what you mean..

the formula you posted was correct.

Originally Posted by the man of science

the formula you posted was correct.

No no!

I'm searching an explanation about that derivative..how to solve it and why..

Originally Posted by doctor_cat

Hi, colleagues!!

Where in my Calculus book do i have to search in order to find an exhaustive explanation of how derivative of a definite integral can be calculated?

Thank you all!

PS: Forgive me for my fool question!

Derivative of a definite integral ? Unless the definite integral has a variable in it somewhere, the integral is a number, a constant, and the derivative is 0.

Are you talking about "differentiating under the integral sign" ?

Perhaps if you defined your terms or at least gave an example we might be able to help your. But I am not sure what you are talking about, and I have taught that sort of class on more than one occasion.

Forgive my defect..I'm sorry!


I'm speaking about:

1. something like
d/dx of , where a = a(x,y) and b= b(x,y);


2. the possibility of doing d/dx before the definite integral is solved;

Originally Posted by doctor_cat

Forgive my defect..I'm sorry!


I'm speaking about:

1. something like
d/dx of , where a = a(x,y) and b= b(x,y);


2. the possibility of doing d/dx before the definite integral is solved;

That is what is generally called differentiating under the integral sign. The way that you have expressed it is a bit confusing, particularly the introduction of "y" into the picture, which doesn't seem to make sense, since that integral is then a function of both x and y and because of that d/dx really should be replaced by a partial derivative.

But the more important complicating aspect is that you have both the function f and the limits of integration depending on x and y. That is a bit unusual.

If you were looking for d/dx , with a and b constant then so long as the partial derivative is continuous it would be .

And if f were a function of one variable only then d/dx .

I don't recall ever seeing a good treatment of differentiating under the integral sign in an elementary calculus book. It is treated in some more advanced books, Lang's Real Analysis being one.

Which calculus text are you using in your class ?

I apologize..in reality I'm not able to write formulas using TeX...

This is what I meant:

with a = a(x,y), b = b(x,y).



PS: I use 'Mathematic Analysis' by Marcellini & Sbordone..I don't know if it is quite well known

Originally Posted by doctor_cat

I apologize..in reality I'm not able to write formulas using TeX...

This is what I meant:

with a = a(x,y), b = b(x,y).



PS: I use 'Mathematic Analysis' by Marcellini & Sbordone..I don't know if it is quite well known

I have never heard of that particular book and I cannot find it at Amazon or Alibris.

Also, I don't know of any general results or approach that apply when both the integrand and the limits of integration are functions of x and y. I suppose that one might be able to do something in special cases.

I think this is what you want:



I could not find my reference so I wrote this down from memory but when ever I use this it makes since.

The first term is how the changing function effects the integral. The second term is how the left boundary effects the integral. If is positive the left boundary is moving in on the integral and decreasing its value, hence the minus sign. We have to multiply by the height of the function at the left boundary to obtain what is lost from the integral hence the . The same logic can be applied to the right boundary.

I'll check with my reference tomorrow to make sure I got this right.

Originally Posted by c186282

I think this is what you want:

Yes, something like that..

today I've found the following formula

http://img523.imageshack.us/my.php?i...9171827zc4.jpg

on a pdf file about a degree thesis (unfortunately, it's written in Italian and I don't give references about it).


@ DrRocket: That's the book I'm using

http://www.hoepli.it/libro.asp?ib=97...ck&from=kelkoo

..I had translated the title so you could not find it on Amazon.

Originally Posted by doctor_cat

Originally Posted by c186282

I think this is what you want:

Yes, something like that..

today I've found the following formula

http://img523.imageshack.us/my.php?i...9171827zc4.jpg

on a pdf file about a degree thesis (unfortunately, it's written in Italian and I don't give references about it).


@ DrRocket: That's the book I'm using

http://www.hoepli.it/libro.asp?ib=97...ck&from=kelkoo

..I had translated the title so you could not find it on Amazon.

I was not correct when I said that I was unaware of any general approach for evaluating integrals of the type that mention. C186282 is basically correct, and I had a bit of a lapse in memory. The relevant theorem is known as Leibniz's Formula and goes like this (this is taken from Elements of Analysis by Bartle):

Liebniz's Formula. Suppose that and are continuous on D to R and that and are functions which are differentiable on the interval [c,d] and have values in [a,b]. If we define on [c,d] by

then has a derivative for each t in[c,d] which is given by


Note tht here is the derivative of at t, etc.

Thank you all!!

PS: So do you suggest me Lang's Real Analysis about integrals?

Originally Posted by doctor_cat

Thank you all!!

PS: So do you suggest me Lang's Real Analysis about integrals?

Lang's Real Analysis is a graduate level text on the subject of Real Analysis. It is a good book, but I would not buy it if your only incentive is the question about integrals.