Riemann Integrals

Overview

Let $f$ be a function defined and bounded on $[a,b]$. Let $s$ and $t$ denote arbitrary step functions defined on $[a,b]$ such that $s(x)\le f(x)\le t(x)$ for all $x\in [a,b]$. If for every such $s$ and $t$, there is exactly one number $I$ satisfying

then $I$ is said to be the Riemann integral of $f$ from $a$ to $b$ and is denoted by symbol ${\int }_a^{b}f(x)dx$. When such an $I$ exists, the function $f$ is said to be integrable on $[a,b]$.

Endpoint $a$ is called the lower limit of integration. Likewise, $b$ is called the upper limit of integration. Together they called the integration limits. $f(x)$ is called the integrand whereas $dx$ is called the differential.

Furthermore, we define

Step Functions

Let $s$ be a step function defined on interval $[a,b]$, and let $P=\{x_0,x_1,\dots ,x_n\}$ be a partition of $[a,b]$ such that $s$ is constant on the open subintervals of $P$. Denote by $s_k$ the constant value that $s$ takes in the $k$th open subinterval, so that for $k=1,2,\dots ,n$,

The Riemann integral of $s$ from $a$ to $b$, denoted by the symbol ${\int }_a^{b}s(x)dx$, is defined by the following formula:

The right-hand side of the above equality is known as a Riemann sum.

Integrability

The lower Riemann integral of $f$, denoted by $\underset{―}{I}(f)$, is defined as

Likewise, the upper Riemann integral of $f$, denoted by $\overline{I}(f)$, is defined as

Thus $f$ is Riemann integrable on $[a,b]$ if and only if ${\int }_a^{b}f(x)dx=\underset{―}{I}(f)=\overline{I}(f)$.

Bounded Monotonic Functions

If $f$ is monotonic on a closed interval $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.

Continuity

Let $f$ be continuous on $[a,b]$. Then $f$ is Riemann integrable on $[a,b]$.

Mean Value Theorem

Let $f$ be continuous on $[a,b]$. Then there exists some $c\in [a,b]$ such that

Indefinite Integrals

Let $f$ be a function such that the integral ${\int }_a^{x}f(t)dt$ exists for each $x$ in an interval $[a,b]$. The function $F$, an indefinite integral of $f$, is given by

If $f$ is integrable on $[a,x]$ for every $x\in [a,b]$, then the indefinite integral $F$ is continuous at each point of $[a,b]$.

Properties

Integrand Additivity

Let $f$ and $g$ be Riemann integrable on $[a,b]$. Then

Vertical Scaling

Let $f$ be Riemann integrable on $[a,b]$ and $c\in \mathbb{R}$. Then

This is also known as the homogeneous property.

Linearity

Let $f$ and $g$ be Riemann integrable on $[a,b]$. Let $c_1,c_2\in \mathbb{R}$. Then

Comparison Theorem

Let $f$ and $b$ be Riemann integrable on $[a,b]$. If $f(x)\le g(x)$ for all $x\in [a,b]$, then

Interval of Integration Additivity

Let $f$ be Riemann integrable over an interval containing $a$, $b$, and $c$. Then

Invariance Under Translation

Let $f$ be Riemann integrable on $[a,b]$ and $c\in \mathbb{R}$. Then

Horizontal Scaling

Let $f$ be Riemann integrable on $[a,b]$. Then for all $k\in \mathbb{R}$ such that $k\ne 0$,

Reflection

The reflection properties are special cases of the vertical and horizontal scaling properties. Let $f$ be Riemann integrable on $[a,b]$. Then

Average Value

Let $f$ be Riemann integrable on an interval $[a,b]$. We define the average value of $f$ on $[a,b]$, denoted $A(f)$, by the formula

Let $w$ be a nonnegative function such that ${\int }_a^{b}w(x)dx\ne 0$. Then the weighted average value of $f$ on $[a,b]$, denoted $A(f)$, is given by formula

In this context, $w$ is called a weight function.

Improper Integrals

An improper integral is an integral that tends to infinity in either dimension. Those of the first kind extends to $\pm \mathrm{\infty }$ over the interval of integration whereas those of the second kind extend over unbounded singularities.

First Kind

Let ${\int }_a^{b}f(x)dx$ exist for all $b\ge a$. An improper integral of the first kind (or infinite integral) is the function $I$ where

The function $I$ is denoted as ${\int }_a^{\mathrm{\infty }}f(x)dx$. It is said to converge if the limit of $I(b)$ as $b\to +\mathrm{\infty }$ exists and is finite. Otherwise it is said to diverge. Integrals of the following form are similarly defined:

If for some $c$ both ${\int }_{-\mathrm{\infty }}^{c}f(x)dx$ and ${\int }_c^{\mathrm{\infty }}f(x)dx$ converge, then we say ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx$ converges and its value is defined as

Second Kind

Let ${\int }_x^{b}f(t)dt$ exists for all $x\in (a,b]$. An improper integral of the second kind is the function $I$ where

The function $I$ is denoted as ${\int }_{a+}^{b}f(t)dt$. It is said to converge if the limit of $I(x)$ as $x\to a+$ exists. Otherwise it is said to diverge. Integrals of the following form are similarly defined:

If for some $c$ both ${\int }_{a+}^{c}f(t)dt$ and ${\int }_c^{b-}f(t)dt$ converge, then we say ${\int }_{a+}^{b-}f(t)dt$ converges and its value is defined as

Linearity

Let $c,\alpha ,\beta \in \mathbb{R}$ and both ${\int }_c^{\mathrm{\infty }}f(x)dx$ and ${\int }_c^{\mathrm{\infty }}g(x)dx$ converge. Then the integral ${\int }_c^{\mathrm{\infty }}(\alpha f(x)+\beta g(x))dx$ converges and its sum is given by equation

Let $c,d,\alpha ,\beta \in \mathbb{R}$ and both ${\int }_{c+}^{d}f(x)dx$ and ${\int }_{c+}^{d}g(x)dx$ converge. Then the integral ${\int }_{c+}^d(\alpha f(x)+\beta g(x))dx$ converges and its sum is given by equation

Monotonicity

Assume the proper integral ${\int }_a^{b}f(x)dx$ exists for each $b\ge a$ and suppose that $f(x)\ge 0$ for all $x\ge a$. Then ${\int }_a^{\mathrm{\infty }}f(x)dx$ converges if and only if there is a constant $M>0$ such that

Assume the proper integral ${\int }_x^{b}f(t)dt$ exists and $f(x)\ge 0$ for each $a<x\le b$. Then ${\int }_{a+}^{b}f(t)dt$ converges if and only if there is a constant $M>0$ such that

Absolute Convergence

Let $c\in \mathbb{R}$. An improper integral ${\int }_c^{\mathrm{\infty }}f(x)dx$ is called absolutely convergent if ${\int }_c^{\mathrm{\infty }}|f(x)|dx$ converges. The series is conditionally convergent if ${\int }_c^{\mathrm{\infty }}f(x)dx$ converges but ${\int }_c^{\mathrm{\infty }}|f(x)|dx$ diverges.

Assume ${\int }_c^{\mathrm{\infty }}|f(x)|dx$ converges. Then ${\int }_c^{\mathrm{\infty }}f(x)dx$ converges, and we have

Let $c,d\in \mathbb{R}$. An improper integral ${\int }_{c+}^{d}f(x)dx$ is called absolutely convergent if ${\int }_{c+}^d|f(x)|dx$ converges. The series is conditionally convergent if ${\int }_{c+}^{d}f(x)dx$ converges but ${\int }_{c+}^d|f(x)|dx$ diverges.

Assume ${\int }_{c+}^d|f(x)|dx$ converges. Then ${\int }_{c+}^{d}f(x)dx$ converges, and we have

Direct Comparison Test

Assume proper integral ${\int }_a^{x}f(t)dt$ exists and that $0\le f(x)\le g(x)$ for all $x\ge a$. If ${\int }_a^{\mathrm{\infty }}g(t)dt$ converges, then ${\int }_a^{\mathrm{\infty }}f(x)dx$ converges and

The integral ${\int }_a^{\mathrm{\infty }}g(t)dt$ is said to dominate the integral ${\int }_a^{\mathrm{\infty }}f(t)dt$.

Assume the proper integral ${\int }_x^{b}f(t)dt$ exists and that $0\le f(x)\le g(x)$ for all $a<x\le b$. If ${\int }_{a+}^{b}g(t)dt$ converges, then ${\int }_{a+}^{b}f(t)dt$ converges and

The integral ${\int }_{a+}^{b}g(t)dt$ is said to dominate the integral ${\int }_{a+}^{b}f(t)dt$.

Limit Comparison Test

Assume both ${\int }_a^{x}f(t)dt$ and ${\int }_a^{x}g(t)dt$ exist, $f(x)\ge 0$, and $g(x)>0$ for all $x\ge a$. If

then ${\int }_a^{\mathrm{\infty }}f(t)dt$ converges if and only if ${\int }_a^{\mathrm{\infty }}g(t)dt$ converges.

Assume both ${\int }_x^{b}f(t)dt$ and ${\int }_x^{b}g(t)dt$ exist, $f(x)\ge 0$, and $g(x)>0$ for all $a<x\le b$. If

then ${\int }_{a+}^{b}f(t)dt$ converges if and only if ${\int }_{a+}^{b}g(t)dt$ converges.