Infinitesimals in integration vs delta x in summations

[MrMultiMedia]

Hi,
I first had a question regarding infinitesimals. What does it mean when the infinitesimal is at the beginning of the integral? For example:

∫dxf(x)
is this the same as
∫f(x)dx ?

My second question was how to convert a summation to an integral and a summation into an integral. Thanks a lot,

-MMM

 

[Fredrik]

Hi,
I first had a question regarding infinitesimals. What does it mean when the infinitesimal is at the beginning of the integral? For example:

∫dxf(x)
is this the same as
∫f(x)dx ?

Yes, those are just two different notations for the same thing.

My second question was how to convert a summation to an integral and a summation into an integral. Thanks a lot,

-MMM

In a physics class (where mathematics is done without rigor), you just do something like this:
$$\sum_i F(x_i) =\sum_i \frac{F(x_i)}{\Delta x_i}\Delta x_i =\sum_i f(x_i)\Delta x_i \approx\int f(x) dx,$$ where $f(x_i)=F(x_i)/\Delta x_i$.

If you want a rigorous answer, I think you will have to make the question more specific.

 

[HallsofIvy]

This may be more picky than necessary, but you are actually asking about the differential, dx. An "infinitesmal" is a completely different matter.