why 'dummy variable' of a definite integral doesn't matter

When we are doing definite integrals, why doesn't the "dummy variable" matter? A dummy variable is a variable we use in a definite integral. We will first see that the dummy variable indeed doesn't matter by using an area argument. Then we will prove that the integral of x^a(1-x)^b from 0 to 1 is the same as the integral of x^b(1-x)^b from 0 to 1 by doing a u-substitution. Hopefully, this video can clear all the questions on the substitution part when we are doing definite integral properties proofs.

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[fanamatakecick]

The hardest part of Calculus is accepting things that seem too good to be true

[mathmathician]

The reason is simple : Unlike indefinite integrals, definite integrals will give us results that does not involved in terms of the variable that will be integrated with respect to (in other words, functions that are not involved in terms of the variable that will be integrated with respect to). That's why, regardless of the variables that will be integrated with respect to, the result of a definite integral is still the same.

[saivivekpeta]

People get confused because they think we are re-substituting the "u substitution". We are not doing that, we just take the u and turn into x without any relation.

[samuelhart]

Awesome video! Derivatives and integrals have been in my mathematical toolkit for quite some time, but one thing that I never understood was why they are miraculously linear functions. It's not intuitive to me why the integral of 1/(1+x^2) + e^-x is actually the sum of the two separate integrals. Perhaps that is a video for another time!

[dfsgjlgsdklgjnmsidrg]

at this point hes just flexing the amount of pens he can hold and draw with

[xinpingdonohoe]

It was strange to me that people didn't think you could just use whatever variables in definite integrals, series, products, etc.. Each one you evaluate is just a number, regardless of what letter you use.

[higherleveltutoring]

Great video!! getting ready for grad school and had to understand why this was the way it was. You hit the nail right on the head, excellent job!

[fivestar]

pretty simple to proof! more such kind of videos, more proofs!

[asriel]

Can you make a video explain wtf is a double integral or triple integral or something like partial derivative?

[Ninja]

The way i understand it is that for definite integrals, the answer is just a number. Regardless of what letter you use, you will get the same number back.

[JoJoDo]

I don't get why it would matter...

[yannstevematex]

I still don't understand because it seems like saying the equation x^m (x+1)^n = x^n(x+1)^m which is false (except for m=n). I tried integrating over the interval (0;3) x/(x+1)² and upon applying what you said I obtained (x-1)/x² but the later function isn't defined at x=0

[cycklist]

Was there anyone at all that didn't understand this???