I'm trying to honestly trying to understand why the Jacobian is present in the change of variable in double integrals and hit a brick wall when trying to do the same for single integrals(ofc u-substitution).
$\int f(x)dx$ and after the u-substitution gives $\int f(g(x))g'(x)dx$ and of course the bounds changed if they were present.
My question is, what is the geometric interpretation of the derivative.
My thinking: Since the substitution in general, changed the original area by stretching or squishing it, this is meant to somehow undo that change but I don't see how a derivative would do that.