Actually I have 2 questions?
1) Prove that the integral of (df/dx) from a to be is equal to f(b) - f(a) using the definition of Riemann Integration and the definition of derivative.
I tried using the antiderivative proof but I dont think that is the right direction on for this one.
2) Prove that the integral of f(x)= x^2 sin(1/x) on the interval (-pi, pi) is beteen -2pi^3/3 and 2pi^3/3.
Bit clueless on this one.
If is an integrable function in the interval and if is a primitive function of in then it verifies that
For any partition of by applyin' the MVT to in each one of intervals we can ensure that it does exist a point (for each ) such that Summing member to member the previous equalities, we get
Let and be the infimum and supremum of in and according to lower and upper sums and of to the partition, respectively, as one wants it it yields
Hence, is included between the lower and upper sums of for any partition, and this property has the integral from here