Definite Integral

Definitions

by Draboux

Darboux Sums

• The Lower Darboux Sum
  • $L(P)=\sum _{i=1}^n{m}_i\cdot \Delta x_i$ where $m_i=\underset{x\in [x_{i-1},x_i]}{\inf }f(x)$
• The Upper Darboux Sum
  • $U(P)=\sum _{i=1}^n{M}_i\cdot \Delta x_i$ where $M_i=\underset{x\in [x_{i-1},x_i]}{\sup }f(x)$
• $L(P)\le U(Q)$ (for any partitions)
• $\underset{P}{\sup }L(P)\le \underset{P}{\inf }U(P)$
• (1.36) $L(P)\le \sigma \le U(P)$

Upper & Lower Darboux integral

• Let $\mathcal{P}$ represent all possible partitions over $[a,b]$

• The upper Darboux integral of $f$ is

  • $U_f=\overline{{\int }_a^b}f(x)\mathrm{d}x=\inf \{U_{f,P}:P\in \mathcal{P}\}$

• The lower Darboux integral of $f$ is

  • $L_f=\underline{{\int }_a^b}f(x)\mathrm{d}x=\sup \{L_{f,P}:P\in \mathcal{P}\}$

• $L_f\le U_f$

• Let $P^{\prime }$ be a refinement of the partition $P$ of the given interval $[a,b]$ then:

  • $U(P^{\prime })\le U(P)$
  • $L(P)\le L(P^{\prime })$

Darboux integral

• (d1.8) A bounded function $f:[a,b]\to \mathbb{R}$ is Darboux-integrable if $\overline{{\int }_a^b}f(x)dx=\underline{{\int }_a^b}f(x)dx$, in such case, the common value is called the Darboux integral of $f$ and denote as ${\int }_a^{b}f(x)dx$

by Riemann

Riemann Sums

Let be a function $f:[a,b]\to \mathbb{R}$ and $P=\{a=x_0,x_1,x_2,\dots ,x_n=b\}$ is a partition of $[a,b]$ and $\Delta x_i=x_i-x_{i-1}$

• (d1.38) $\mid \mid P\mid \mid =\lambda (P)=\underset{1\le i\le n}{\max }\Delta x_i$ is the mesh (or norm) of $P$
• (d1.35) The sum $\sigma =R(P)=\sum _{i=1}^{n}f({\xi }_i)\cdot \Delta x_i$ where ${\xi }_i$ is a chosen point in the $i$-th subinterval $[x_{i-1},x_i]$ is called a Riemann sum of $f$ over $[a,b]$ with partition $P$

Riemann Integral

• (d1.39) ${\int }_a^{b}f(x)dx=\underset{\lambda (P)\to 0}{\lim }\sigma =\underset{\mid \mid P\mid \mid \to 0}{\lim }\sum _{i=1}^{n}f({\xi }_i)\cdot \Delta x_i$ is the definite integral of $f$ from $a$ to $b$.

  • if the limit exists (finite) then $f$ is integrable
  • $\forall \epsilon >0,\exists \delta >0:\left(\underset{1\le i\le n}{\max }\Delta x_i<\delta ⟹\left|(\sum _{i=1}^{n}f({\xi }_i)\cdot \Delta x_i)-{\int }_a^{b}f(x)dx\right|<\epsilon \right)$

• (1.40) if $f$ is Riemann-integrable on $[a,b]$ then $f$ is bounded on $[a,b]$

draft --- other def

• Consider the partition $P_n=\{a=x_0,x_1,x_2,\dots ,x_n=b\}$ of $[a,b]$ such that the subintervals have equal width $\Delta x=(b-a)/n$.
• A Riemann sum of $f$ over $[a,b]$ with partition $P_n$ is given by $\sum _{i=1}^{n}f({\xi }_i)\cdot \Delta x$, where ${\xi }_i$ is a chosen point in the $i$-th subinterval $[x_{i-1},x_i]$.
• The Riemann integral of $f(x)$ over $[a,b]$, denoted by ${\int }_a^{b}f(x)dx$, is defined as ${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f({\xi }_i)\cdot \Delta x$.

Riemann & Darboux Equivalence

Equivalence of Definitions of Riemann and Darboux Integrals

• (1.42) $f$ is Riemann-integrable iff $f$ Darboux-integrable, in this case Riemann-integral is equal to Darboux-integral

(half)-open interval

• (u3) Given $f$ is a function defined and bounded on $(a,b]$. Let $f_1(x)=\{\begin{array}{ll}f(x)& x\in (a,b]\\ c& x=a\end{array}$ where $c$ is any constant. Then $f$ is integrable on $(a,b]$ if and only if $f_1$ is integrable on $[a,b]$. In this case ${\int }_a^b{f}_1={\int }_a^{b}f$. (similar definitions for $[a,b)$ and $(a,b)$).
• (q3.2) A function $f$ (defined and bounded on $(a,b]$) is not integrable on $(a,b]$ if and only if there exists a function $f_1$ (defined as above) that is not integrable on $[a,b]$.

Area Function

Given $f$ is an function integrable on $[a,b]$

• The function $F_c(x)={\int }_c^{x}f(t)dt$ is called an area function of $f$ on $[a,b]$ (where $c\in [a,b]$), (אינטגרל בלתי מסוים (d1.30))
• Given the area function $F_c(x)$ of $f$ on $[a,b]$ (where $c\in [a,b]$)
  • The area function $F_c(x)$ gives the net area of the region bounded by the graph of $f$ and the $t$-axis on the interval $[c,x]$
  • (q1.57) $F_c(x)$ is an antiderivative of $f$ on $[a,b]$
  • (1.32) $F_c(x)$ is continuous on $[c,b]$
  • (1.33) if $f$ is continuous on $[a,b]$ then $F_c(x)$ differentiable on $[c,b]$, and $F_c^{\prime }(x)=f(x)$
• Given the area functions $F_c$ and $F_d$ of $f$ on $[a,b]$ (where $c,d\in [a,b]$)
  • (1.31) There exists a constant $K$ such that $F_c(x)-F_d(x)=K$. (also $K={\int }_c^{d}f(t)dt$)

Theorems

• (1.10) $f$ is integrable, if and only if, $\forall \epsilon >0,\exists P:U_{f,P}-L_{f,P}<\epsilon$

• (1.11) $\underset{n\to \infty }{\lim }(U(P_n)-L(P_n))=0⟹{\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }U(P_n)=\underset{n\to \infty }{\lim }L(P_n)$

• (1.20,q1.34) Additivity

  • If $a<c<b$ then $f$ is integrable on $[a,b]$ iff $f$ is ingerable on $[a,c]$ and $[c,b]$. In that case: ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$

• (1.25) If $f(x)=g(x)$ for every $x$ excluding finite number of points on $[a,b]$, then, if $g$ integrable on $[a,b]$ then $f$ integrable on $[a,b]$, and ${\int }_a^{b}f(x)dx={\int }_a^{b}g(x)dx$

• if $f$ is integrable on an interval where $a$ and $b$ are its endpoints and $|f(x)|\le M$ for all $x$ in that interval, then $\left|{\int }_a^{b}f(t)dt\right|\le M|b-a|$

• (1.41) if $f$ is Darboux-integrable on $[a,b]$, then $\underset{\lambda (P)\to 0}{\lim }(S(P)-s(P))=0$

• Odd / Even Functions (continuous on $\mathbb{R}$)

  • (q2.43, Odd) $f(-x)=-f(x)$ for all $x$, then ${\int }_{-a}^{a}f(x)dx=0$ for all $a$.
  • (Even) if $f(-x)=f(x)$ for all $x$, then ${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$ for all $a$.

• Periodic functions

  • If $f(x)=f(x+P)$ for all $x$, then ${\int }_{a+P}^{b+P}f(x)dx={\int }_a^{b}f(x)dx$ for all $a,b$.

Integrabletly Sufficient Conditions

Sufficient conditions for integrabletly of $f$ on $[a,b]$:

• (1.11) $\underset{n\to \infty }{\lim }(U(P_n)-L(P_n))=0$
• (1.15) if $f$ is weakly monotonic on $[a,b]$
• (1.17) if $f$ is bounded and piecewise monotone on $[a,b]$
• (1.18) if $f$ is continuous (thus bounded) on $[a,b]$
• (1.19) if $f$ is bounded and continuous on $[a,b]$ (possibly expect finites number of discontinuity points)
• if $f$ is piecewise continuous function on $[a,b]$

Integrable Function Properties

Given $f$ (and $g$) is integrable (usually on $[a,b]$, unless otherwise stated)

• (1.13, הנוסחה היסודית, Newton–Leibniz theorem, 2nd fundamental theorem of calculus) - If $F$ is an antiderivative of $f$ on $[a,b]$, then:
  • ${\int }_a^{b}f(x)dx=F(b)-F(a)=F(x){|}_a^b$
• $\forall a,b,c$ we have ${\int }_a^{b}cdt=c(b-a)$
• (1.21) $f$ is integrable on $[c,d]\subseteq [a,b]$
• (d1.22a) Reversing Limits: ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
• (d1.22b) Identical Limits (Zero Width Interval): ${\int }_a^{a}f(x)dx=0$
• Linearity
  • (1.24a) Constant Multiple
    • ${\int }_a^{b}cf(x)dx=c{\int }_a^{b}f(x)dx$
      • ($c=-1$ special-case) ${\int }_a^b(-f(x))dx=-{\int }_a^{b}f(x)dx$
  • (1.24b) Sum
    • ${\int }_a^b(f(x)\pm g(x))dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx$
• (1.23) Additivity: ${\int }_a^{b}f(x)dx={\int }_a^{c}f(x)dx+{\int }_c^{b}f(x)dx$ (given $a,b,c\in \mathbb{R}$, and $f$ is integrable on each closed intervals that their endpoints are two of $a,b,c$)
• (1.26) Monotonicity
  • $\forall x\in [a,b],f(x)\le g(x)⟹{\int }_a^{b}f(x)dx\le {\int }_a^{b}g(x)dx$
• (1.27) $|f|$ is also integrable on $[a,b]$
• (1.32) $F_a(x)={\int }_a^{x}f(t)dt$ is continuous on $[a,b]$
• (1.31) ${\int }_c^{x}f(t)dt-{\int }_d^{x}f(t)dt={\int }_c^{d}f(t)dt$ for all $c,d$. (i.e. $F_c(x)-F_d(x)={\int }_c^{d}f(t)dt$)
• (1.34) if $f$ has an antiderivative on $[a,b]$ then $F_c(x)={\int }_c^{x}f(t)dt$ is an antiderivative of $f$ on $[a,b]$
• (q1.50b) $\left|{\int }_a^{b}f(x)dx\right|\le {\int }_a^b|f(x)|dx$
• (q1.51) Max-Min Inequality - if $m\le f(x)\le M$ for all $x\in [a,b]$ then $m(b-a)\le {\int }_a^{b}f(x)dx\le M(b-a)$
• $\forall x\in [a,b],f(x)\ge 0⟹{\int }_a^{b}f(x)dx\ge 0$

Continuous Function Properties

if $f$ is continuous on $[a,b]$ then:

• (1.18) $f$ is integrable on $[a,b]$ (thus $f$ has integrable function properties)
• (1.29) MVT for Integrals
  • $\exists c\in [a,b]:{\int }_a^{b}f(x)dx=f(c)(b-a)$
  • (see also Average Value)
• The Fundamental Theorem of Calculus
  • (1.33)
    • The area function $F_a(x)={\int }_a^{x}f(t)dt$ is differentiable on $[a,b]$, and
    • $F_a^{\prime }(x)=\frac{d}{dx}\left[{\int }_a^{x}f(t)dt\right]=f(x)$
  • (1.33’)
    • $f$ has an antiderivative function on $[a,b]$
    • every antiderivative function of $f$ on $[a,b]$ is of the form ${\int }_a^{x}f(t)dt+C$ where $C$ is some real number
  • (see also Newton–Leibniz theorem which is sometimes referred to as the second fundamental theorem of calculus)
• (q1.57) for all $c\in [a,b]$, the area function $F_c(x)={\int }_c^{x}f(t)dt$ is an antiderivative of $f$ on $[a,b]$

Applications

• The Net Change Theorem - The net change in a differentiable function $F(x)$ over $[a,b]$ is is the integral of its rate of change, which is ${\int }_a^b{F}^{\prime }(x)dx=F(b)-F(a)$

The Area Problem

A procedure for finding areas via integration is called the antiderivative method

• If $f$ (and $g$) are continuous

  • ${\int }_a^b|f(x)|dx$ is the area of of the region that lies between the graph of $f$ and the interval $[a,b]$ on the $x$-axis
  • $A=\left|{\int }_a^b(f(x)-g(x))dx\right|$ is the area between the curves $f(x)$ and $g(x)$ over the interval $[a,b]$

• If $f(x)$ is nonnegative and integrable over a closed interval $[a,b]$, then the area under the curve $f(x)$ over $[a,b]$ is the integral of $f$ from $a$ to $b$, $A={\int }_a^{b}f(x)dx$

Average Value

• If $f$ is integrable on $[a,b]$, then its average value on $[a,b]$ is $\frac{1}{b-a}{\int }_a^{b}f(x)dx$ (see also MVT for Integrals)

Arc Length

• If $f^{\prime }$ is continuously differentiable on $[a,b]$, then the length (arc length) of the curve $y=f(x)$ from the point $(a,f(a))$ to the point $(b,f(b))$ is $L={\int }_a^b\sqrt{1+(f^{\prime }(x){)}^2}dx={\int }_a^b\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx$

Volume

Solid of Revolution

The solid generated by rotating (or revolving) a planar region about an axis in its plane is called a solid of revolution.

• The Disk Method

  • Rotation about the $x$-axis
    • $R(x)$ is the distance from the $x$-axis of revolution to the planar region’s boundary (the curve $y=R(x)$)
    • $A(x)=\pi \cdot (R(x){)}^2$ is the area of a disk of radius $R(x)$
    • $V={\int }_a^{b}A(x)dx$ is the volume of the solid generated by revolving a region between the $x$-axis and a curve $y=R(x)$ (where $a\le x\le b$) about the $x$-axis
  • Rotation about the $y$-axis
    • $R(y)$ is the distance from the $y$-axis of revolution to the planar region’s boundary (the curve $x=R(y)$)
    • $A(y)=\pi (R(y){)}^2$ is the area of a disk of radius $R(y)$
    • $V={\int }_c^{d}A(y)dy$ is the volume of the solid generated by revolving a region between the $y$-axis and a curve $x=R(y)$ (where $c\le y\le d$) about the $y$-axis

• The Washer Method