# Question on integral notation (dt, dx, etc.)

1. May 9, 2010

### hulgy

Say the limit of the integral is from 0 to t and the integral is ended with a dt. Is this okay?

Generally, all the integrals I see with a variable limit end with a d-letter that is not the same as the variable in the limit. ie: limit is from 0 to t, ends with du

2. May 9, 2010

### tiny-tim

Welcome to PF!

Hi hulgy! Welcome to PF!
Noooo …

If you have an ∫ f(t) dt, that has to be between limits of t …

t can't depend on itself.

(once you've integrated over a variable like t, that variable should disappear, not come back to haunt you!)

3. May 9, 2010

### hulgy

Re: Welcome to PF!

Hmmm... what if t stood for a value like 1<t<3 which you would have to place into the 0 to t limit (this would technically eliminate the variable...right?). Would the integral as a whole still be wrong then?

4. May 9, 2010

### tiny-tim

Not following you

can you write the integral out in full?

(you can write integrals like this: [noparse]∫ab[/noparse] … ∫ab )

5. May 9, 2010

### hulgy

0t (50)dt for 1<t<3

6. May 9, 2010

### tiny-tim

hmm …

If you mean ∫0s (50)dt for 1<s<3, then that's fine.

But no, you can't use t for two different things.

7. May 9, 2010

### D H

Staff Emeritus
That is a bit of abuse of notation -- not that you won't see such abuse.

The dt inside the integral is a dummy variable. It might as well be tau, or x, or anything. It has nothing to do with the upper limit of the integration.

This is at best confusing:

$$f(t) = \int_0^t \cos t\, dt$$

What is $df/dt$?

If the above integral is expressed as

$$f(t) = \int_0^t \cos \tau\,d\tau$$

then it is pretty clear that $f(t)=\sin t$ and that $df/dt = \cos t$.

Occasionally you will run across things like

$$f(t) = \int_0^t \cos (t-\tau)\,d\tau$$

The t inside the above integral is not a dummy variable. As far as the integration is concerned that t inside the integral is a constant.

8. May 9, 2010

### hulgy

I guess I'm gonna get points off for notation on my AP calc test then. Well thanks for all the help, really cleared things up.