Approximations of the Riemann Integral in R

Subjects: Real Analysis
Links: Riemann Integral in R

Equal Partitions

Let ${\mathcal{P}}_n$ be uniform partitions of $n$ elements. Evaluating the Reimann sums at left or right endpoints is an important construction. They receive a particular notation, where $h_n=\frac{b-a}{n}$:

If we don’t have a reason to prefer either one of them, it’s reasonable to take the mean we get:

In particular, if $f$ is monotonic then the integral will be trapped between the two, and we will get that the average will differ of the integral by a small amount:

Trapezoidal Rule

Continue to work with equal partitions, ${\mathcal{P}}_{\mathcal{n}}$ and a width of $h_n=\frac{b-a}{n}$ . Let’s consider the function $g_n$ is a piecewise function, of a linear approximation of $f$ at the interval $[a+(k-1)h_n,a+k{h}_n]=I_k$. In other words:

Then the integral along all $[a,b]$ will be equal to:

$T_n(f)$ is the n-th Trapezoidal Approximation.

Theorem: If $f\in {\mathcal{C}}^{\mathcal{2}}[a,b]$, then there’s a $c\in [a,b]$:

Corollary: If $f\in {\mathcal{C}}^{\mathcal{2}}[a,b]$ and let $|f^{″}|\le B_2$, then:

Midpoint Rule

Theorem: if $f\in {\mathcal{C}}^{\mathcal{2}}\mathcal{[}\mathcal{a}\mathcal{,}\mathcal{b}\mathcal{]}$, then there’s a $c\in [a,b]$ that:

Corollary: If $f\in {\mathcal{C}}^{\mathcal{2}}[a,b]$ and let $|f^{″}|\le B_2$, then:

How to transform knowledge of $T_n(f)$ and $M_n(f)$ into a type of $T_n(f)$:

Simpson’s Rule

Given a $n\in \mathbb{2}\mathbb{N}$, the Simpson Sum is defined as follows:

Theorem: If $f\in {\mathcal{C}}^{\mathcal{4}}[a,b]$, there’s a $c\in [a,b]$ that:

Corollary: If $f\in {\mathcal{C}}^{\mathcal{4}}[a,b]$, and $|f^{(4)}|\le B_4$, then:

How to transform knowledge of $T_n(f)$ and $M_n(f)$ into a type of $S_n(f)$: