Earlier today, I asked a question about a proof using the Riemann-sums:
I'm reading a script right now, in which $$\int_{a}^{b} e^x dx = e^b-e^a$$ is proven using Riemann-sums. Unfortunately, I have trouble understanding a step in the beginning.
Some background: The subdivision of an interval $[a,b] \subset \mathbb{R}$ is put as: $[a,b] = [x_0,x_1]\cup [x_1,x_2]\cup...\cup[x_{m-1},x_m]$ with $a = x_0<x_1<x_2<...<x_{m-1}<x_m = b$.
The "fineness" (I'm not sure this is the right translation, could also be "acuteness") is:
$d(\{x_i\}) =$ max$|x_{j-1}-x_j|=$ max$\triangle x_j$.
The fulcrum in every interval $[x_{j-1}-x_j]$ is called $\xi_j$, and $f(\xi_j)$ is the value of the fulcrum.
Finally, the Riemman-sum is definded as: $$\rho(f;\{x_j\},\{\xi_j\}) = \sum_{j=1}^{m} f(\xi_j)\triangle x_j = \sum_{j=1}^{m} f(\xi_j)(x_j-x_{j-1})$$.
Now, the step I can't understand is: $$\sum_{j=1}^{m} e^{\xi_j}\triangle x_j = e^b-e^a+\sum_{j=1}^{m}(e^{\xi_j}-\frac{e^{x_j}-e^{x_{j-1}}}{x_j-x_{j-1}})\triangle x_j$$,
whereas the first step I can still follow. Before, we had taken the Riemann-sum of $f(x)=x$, in which the same expanding step felt intuitive. So if anybody has a hint on how that step is legitimate or on where I could read more about that proof, I'd be very thankful.
to which I got some really good answers.
Now that I have reviewed the proof, I have some further questions. The proof continues by stating:
$= e^b-e^a+\sum_{j=1}^{m}(e^{\xi_j}-e^{\eta_j})\triangle x_j$ mit $\xi_j, \eta_j \in [x_{j-1}, x_j]$.
Which I understand to be an application of the mean value theorem.
But then, it continues by stating:
"such that:
$e^{\xi_j}-e^{\eta_j}=e^{\zeta_j}(\xi_j-\eta_j)$ mit $\zeta \in [x_{j-1},x_j]$,
which is used to conclude
$|\rho-I| \leq \sum_{j=1}^{m}e^b \triangle x_j \cdot \triangle x_j \leq e^b \cdot \delta \cdot(b-a) \leq \epsilon$
(earlier, we set $I$ to be $$I=\int_{a}^{b}e^x$$)
as long as:
$\delta \leq \frac{\epsilon}{e^b(b-a)}$.
Could anybody provide any further background, not concerning the epsilon-delta proof in general, which I understand, but the general way of deduction, which I can't grasp, yet.
To find $\zeta_j$, just apply the mean value theorem again, for $\xi_j$ and $\eta_j$.
Note that $e^x$ is continuous and bounded on $[a,b]$, hence it is Riemann integrable.
Then the Riemann sums must converge to the integral for any internal $\xi_j$ values, as the subdivision gets infinitely fine.
In particular, we can choose $\xi_j:=\eta_j$, so that all the Riemann sums become $e^b-e^a$.
Note also that the same method applies for any continuous function $f$ with an antiderivative $F$ (so that $F'=f$), yielding that $$\int_a^bf=F(b) - F(a)\,. $$