i did a little proof of this for fun, but i just wanna know if anyone else who has taken calc III has made the connection between the mean value theorem for definite integrals and change of varibles with jacobinans. i was like.. hmm.. area between curves as a rectangle--> mvt for DI. just let one v curve be 0, and one f(x*) (that value you solve for as the height of your rectangle in the mvt) and one u curve be a, and the other b. then evaluate the double integral with the jacobian or whatever. anyone had this idea?
valohim6 Wrote:i did a little proof of this for fun, but i just wanna know if anyone else who has taken calc III has made the connection between the mean value theorem for definite integrals and change of varibles with jacobinans. i was like.. hmm.. area between curves as a rectangle--> mvt for DI. just let one v curve be 0, and one f(x*) (that value you solve for as the height of your rectangle in the mvt) and one u curve be a, and the other b. then evaluate the double integral with the jacobian or whatever. anyone had this idea?
To decide whether I can say something to this problem I first have to exactly understand what you mean. Would you please reformulate you problem using some more of these symbols mathematicians like so much :-)?
Hm, I also didnt quite understand the claim.
We have the mean value theorem, that means that the area below f (from a to b) is equal to the area of a rectangle (with width b-a and) with height
)
for some
in the interval (a,b).
 dx = (b-a) f ( x^\ast ) )
And now I think valohim considers the area below f given as bounded by 4 curves, i.e. (t,0), (a,t), (b,t) and (t,f(t)). And to compute the area, he somehow wants to use a double integral and apply change of variables?
Would be nice if you could explain it, valohim.