Improper Integrals

Riemann-Sums

Definition: Riemann Sum

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function, let $I=[a;b]\subseteq \mathcal{D}$ be a closed interval and let $x_0<x_1<\cdots <x_n\in I$.

A Riemann sum of $f$ over $I$ is any sum of the form

$$\sum _{i=1}^{n}f(x_i^{\ast })\Delta x_i,$$

where $\Delta x_i=(x_i-x_{i-1})$ and $x_i^{\ast }\in [x_{i-1};x_i]$.

Note: Choice of $x_i^{\ast }$

Different choices for $x_i^{\ast }$ yield different Riemann sums.

Definition: Left Riemann Sum

A left Riemann sum has $x_i^{\ast }=x_{i-1}$ for all $i$.

Definition: Right Riemann Sum

A right Riemann sum has $x_i^{\ast }=x_i$ for all $i$.

Definition: Left Riemann Sum

A middle Riemann sum has $x_i^{\ast }=\frac{x_{i-1}+x_i}{2}$ for all $i$.

Definite Integrals

Definition: Riemann-Integrability

A real function $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ is Riemann-integrable on the interval $I=[a;b]\subseteq \mathcal{D}$ iff all of its Riemann sums on $I$ have the same limit as $\Delta x_i$ approaches $0$.

$$\underset{\Delta x_i\to 0}{\lim }\sum _{i=1}^{n}f(x_i^{\ast })\Delta x_i=S\in \mathbb{R}$$

Definition: Definite Integral

If the aforementioned limit exists, then we call it the definite integral of $f$ over $I$.

NOTATION

The most common notation for the definite integral is

$${\int }_a^{b}f(x)\mathrm{dx}$$

The upside of this notation is that it clearly shows where the integrand, i.e. the thing being integrated, begins and where it ends. This is not very useful when we are referring to $f$ by its name, but it helps to remove ambiguity when we substitute $f$ with an expression such as ${\int }_a^b{x}^3+\ln x\mathrm{dx}$.

Its main downside is that it forces us to assign a symbol to the function’s argument which is redundant and can even be confusing in some contexts where we refer to $f$ by its name. In particular, it is irrelevant whether we write ${\int }_a^{b}f(x)\mathrm{dx}$ or ${\int }_a^{b}f(y)\mathrm{dy}$ or ${\int }_a^{b}f(\omega )\mathrm{d\omega }$, hence we can shorten the notation to just

$${\int }_a^{b}f$$

The main downside of this notation is that it implies that the order of $a$ and $b$ matters, which is not the case - what matters is the actual set which $[a;b]$ represents. To emphasise this, we can use the following notations:

$${\int }_{[a;b]}f{\int }_{D}f$$

In the latter case, we can also add $\mathrm{d}D$ to clarify where the integrand begins and ends, such as

$${\int }_{D}f\mathrm{dD}$$

All of these notations are useful in specific contexts and less so in others.

Notation: Definite Integrals with Special Bounds

A common convention is to define the notations

$${\int }_b^{a}f=-{\int }_a^{b}f$$

and

$${\int }_c^{c}f=0$$

for each $c\in [a;b]$. This is merely notation which makes the formulation of many theorems easier and more natural.

Note: Definite Integrals over Non-Interval Domains

If $I$ is not an interval but can be represented as the union of finitely many closed intervals $I_1\cup \cdots \cup I_n$ such that each interval overlaps with at most two other intervals and only at the endpoints, then we define the definite integral of $f$ over $I$ as the sum of $f$‘s definite integrals over $I_1,\dots ,I_n$:

$${\int }_{I}f={\int }_{I_1}f\mathrm{d}{I}_1+\cdots +{\int }_{I_n}f\mathrm{d}{I}_n$$

Properties

Theorem: Linearity of the Definite Integral

The definite integral is linear - if $f$ and $g$ are real functions which are Riemann-integrable on $I$, then for all $\alpha ,\beta \in \mathbb{R}$

$${\int }_I\alpha f(x)+\beta g(x)\mathrm{dx}=\alpha {\int }_{I}f(x)\mathrm{dx}+\beta {\int }_{I}g(x)\mathrm{dx}$$

PROOF

TODO

Theorem: Integration by Parts

Let $f$ and $g$ are real functions which are defined on the closed interval $[a;b]$ and are continuously differentiable on the open interval $(a;b)$, then they are Riemann-integrable on $[a;b]$ with

$${\int }_a^{b}f(x)g^{\prime }(x)\mathrm{dx}=f(b)g(b)-f(a)g(a)-{\int }_a^b{f}^{\prime }(x)g(x)\mathrm{dx}$$

PROOF

TODO

Theorem: Substitution

Let $g:[a;b]\to \mathbb{R}$ and $f:[c;d]\to \mathbb{R}$ be real functions such that the image of $g$ is a subset of $[c;d]$, i.e. $g([a;b])\subseteq [c;d]$.

If $f$ is continuous and $g$ is continuously differentiable, then the following definite Integral can be solved via substitution.

$${\int }_a^{b}f(g(x))g^{\prime }(x)\mathrm{dx}={\int }_{g(a)}^{g(b)}f(u)\mathrm{du}$$

PROOF

TODO

Mean Value Theorem for Definite Integrals

Let $f,g:D\to \mathbb{R}$ be real functions on a closed interval $D=[a;b]\subset \mathbb{R}$.

If $f$ and $g$ are continuous and $g(x)>0$ for all $x\in D$, then there exists some $\xi \in (a;b)$ such that

$${\int }_a^{b}f(x)g(x)\mathrm{dx}=f(\xi ){\int }_a^{b}g(x)\mathrm{dx}$$

PROOF

TODO

Tip: $g(x)=1$

In the case of $g(x)=1$ for all $x$, the mean value theorem simplifies to

$${\int }_a^{b}f(x)\mathrm{dx}=f(\xi )(b-a)$$

Theorem: Integrability of the Absolute Value

Let $f$ be a real function.

If $f$ is Riemann-integrable on $l$, then so is its absolute value $|f|$ and

$$\left|{\int }_{I}f(x)\mathrm{dx}\right|\le {\int }_I|f(x)|\mathrm{dx}$$

PROOF

TODO

Improper Integrals

Notation: Improper Integrals

An improper integral is a definite integral for a real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ on an open or a semi-open interval $D$:

• If $D=[a;b)$ where $a,b\in \mathbb{R}$, the improper integral is defined through the left-sided limit

${\int }_a^{b}f(x)\mathrm{dx}\stackrel{\text{def}}{=}{\lim }_{\beta \to b^{-}}{\int }_a^{\beta }f(x)\mathrm{dx}$

• If $D=(a;b]$ where $a,b\in \mathbb{R}$, the improper integral is defined through the right-sided limit

${\int }_a^{b}f(x)\mathrm{dx}\stackrel{\text{def}}{=}{\lim }_{\alpha \to a^{+}}{\int }_{\alpha }^{b}f(x)\mathrm{dx}$

• If $D=[a;\infty )$ where $a\in \mathbb{R}$, the improper integral is defined through the limit

${\int }_a^{\infty }f(x)\mathrm{dx}\stackrel{\text{def}}{=}{\lim }_{\beta \to \infty }{\int }_a^{\beta }f(x)\mathrm{dx}$

• If $D=(-\infty ;b]$ where $b\in \mathbb{R}$, the improper integral is defined through the limit

${\int }_{-\infty }^{b}f(x)\mathrm{dx}\stackrel{\text{def}}{=}{\lim }_{\alpha \to -\infty }{\int }_{\alpha }^{b}f(x)\mathrm{dx}$

• If $D=(-\infty ;\infty )$, then for any choice of $c\in \mathbb{R}$ the improper integral is defined as

${\int }_{-\infty }^{\infty }f(x)\mathrm{dx}\stackrel{\text{def}}{=}{\int }_{-\infty }^{c}f(x)\mathrm{dx}+{\int }_c^{\infty }f(x)\mathrm{dx}$

If the respective limit exists, then the improper integral is said to converge or simply exist. Otherwise, it diverges or does not exist.

Convergence Criteria

THEOREM

Let $f,g:[a;\infty )\to \mathbb{R}$ be Riemann-integrable on every closed interval $[a;b]$ where $a\le b$.

If $|f(x)|\le g(x)$ for all $x\in [a;\infty )$ and the integral ${\int }_a^{\infty }g(x)\mathrm{dx}$ converges, then ${\int }_a^{\infty }f(x)\mathrm{dx}$ also converges.

PROOF

TODO

THEOREM

Let $f,g:(-\infty ;b]\to \mathbb{R}$ be Riemann-integrable on every closed interval $[a;b]$ where $a\le b$.

If $|f(x)|\le g(x)$ $x\in [a;\infty )$ the integral ${\int }_{-\infty }^{b}g(x)\mathrm{dx}$ converges, then ${\int }_{-\infty }^{b}f(x)\mathrm{dx}$ also converges.

PROOF

TODO

Bibliography