What's wrong with this notation? (1 Viewer)

[Trebla]

The definite integral ∫f(x) dx (from a to x) is considered 'sloppy' notation and instead we replace the x in the integrand with a dummy variable, say t, and we get ∫f(t) dt (from a to x), which of course supposedly gives the same answer assuming f(x) is continuous.

So why is ∫f(x) dx (from a to x) supposedly an incorrect notation, even though it works regardless of what variable we set the integrand to be?

 

[Trebla]

Both the 'x' in the integrand and the 'x' in a boundary are variables. Integrals can be expressed over a constant 'a' to a variable 'x' can't they?

 

[darkliight]

Not really. Yes you can choose the 'x' in the boundary to be anything you like, but the key thing is you must choose it to evaluate the integral - essentially a constant. The t in the integrand does not need to be fixed, in fact it can't be. It's a variable that must take on all values from a to x.

 

[LoneShadow]

The definite integral ∫f(x) dx (from a to x) is considered 'sloppy' notation and instead we replace the x in the integrand with a dummy variable, say t, and we get ∫f(t) dt (from a to x), which of course supposedly gives the same answer assuming f(x) is continuous.

So why is ∫f(x) dx (from a to x) supposedly an incorrect notation, even though it works regardless of what variable we set the integrand to be?

don't question the gods of mathematics puny first year kiddo........ you'lll get used to it!

 

[Slidey]

You don't tend to give very good advice, LoneShadow.

 

[Captain Gh3y]

The definite integral ∫f(x) dx (from a to x) is considered 'sloppy' notation and instead we replace the x in the integrand with a dummy variable, say t, and we get ∫f(t) dt (from a to x), which of course supposedly gives the same answer assuming f(x) is continuous.

So why is ∫f(x) dx (from a to x) supposedly an incorrect notation, even though it works regardless of what variable we set the integrand to be?

Integrate f(x) dx from a to x

let x = 5 say

Integrate f(5) d5 from a to 5

YOU CAN'T INTEGRATE WITH RESPECT TO 5!

 

[LoneShadow]

You don't tend to give very good advice, LoneShadow.

I do. Honest. You just have to take a transform of what I say to see the true message

 

[kony]

YOU CAN'T INTEGRATE WITH RESPECT TO 5!

d(5)/dx = 0

therefore, d(5) = 0


therefore, I = 0

 

[Captain Gh3y]

It doesn't work that way

EDIT: Because that's 5 as a function, not a variable

 

[kony]

hmm fair enough.

does the Reimann integral restrict d(x) for x being a variable function?