Add properties of the integral of a step function.
parent
68d46e1a7d
commit
43dd9c2997
|
|
@ -31,7 +31,7 @@ The \textbf{characteristic function} of $S$ is the function $\mathcal{X}_S$ such
|
|||
\begin{definition}
|
||||
|
||||
\lean{Common/Set/Basic}{Set.characteristic}
|
||||
|
||||
|
||||
\end{definition}
|
||||
|
||||
\section{\defined{Infimum}}%
|
||||
|
|
@ -51,10 +51,10 @@ Such a number $B$ is also known as the \textbf{greatest lower bound}.
|
|||
|
||||
\end{definition}
|
||||
|
||||
\section{\partial{Integral of Step Function}}%
|
||||
\section{\partial{Integral of a Step Function}}%
|
||||
\label{sec:def-integral-step-function}
|
||||
|
||||
Lset $s$ be a \nameref{sec:def-step-function} defined on $[a, b]$, and let
|
||||
Let $s$ be a \nameref{sec:def-step-function} defined on $[a, b]$, and let
|
||||
$P = \{x_0, x_1, \ldots, x_n\}$ be a \nameref{sec:def-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,
|
||||
|
|
@ -576,7 +576,7 @@ For each set $S$ in $\mathscr{M}$, we have $a(S) \geq 0$.
|
|||
\end{axiom}
|
||||
|
||||
\section{\defined{Additive Property}}%
|
||||
\label{sec:additive-property}
|
||||
\label{sec:area-additive-property}
|
||||
|
||||
If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
|
||||
$\mathscr{M}$, and we have $a(S \cup T) = a(S) + a(T) - a(S \cap T)$.
|
||||
|
|
@ -589,7 +589,7 @@ If $S$ and $T$ are in $\mathscr{M}$, then $S \cup T$ and $S \cap T$ are in
|
|||
\end{axiom}
|
||||
|
||||
\section{\defined{Difference Property}}%
|
||||
\label{sec:difference-property}
|
||||
\label{sec:area-difference-property}
|
||||
|
||||
If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
|
||||
$\mathscr{M}$, and we have $a(T - S) = a(T) - a(S)$.
|
||||
|
|
@ -602,7 +602,7 @@ If $S$ and $T$ are in $\mathscr{M}$ with $S \subseteq T$, then $T - S$ is in
|
|||
\end{axiom}
|
||||
|
||||
\section{\defined{Invariance Under Congruence}}%
|
||||
\label{sec:invariance-under-congruence}
|
||||
\label{sec:area-invariance-under-congruence}
|
||||
|
||||
If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
|
||||
also in $\mathscr{M}$ and we have $a(S) = a(T)$.
|
||||
|
|
@ -615,7 +615,7 @@ If a set $S$ is in $\mathscr{M}$ and if $T$ is congruent to $S$, then $T$ is
|
|||
\end{axiom}
|
||||
|
||||
\section{\defined{Choice of Scale}}%
|
||||
\label{sec:choice-scale}
|
||||
\label{sec:area-choice-scale}
|
||||
|
||||
Every rectangle $R$ is in $\mathscr{M}$.
|
||||
If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
|
||||
|
|
@ -628,7 +628,7 @@ If the edges of $R$ have lengths $h$ and $k$, then $a(R) = hk$.
|
|||
\end{axiom}
|
||||
|
||||
\section{\partial{Exhaustion Property}}%
|
||||
\label{sec:exhaustion-property}
|
||||
\label{sec:area-exhaustion-property}
|
||||
|
||||
Let $Q$ be a set that can be enclosed between two step regions $S$ and $T$, so
|
||||
that
|
||||
|
|
@ -664,8 +664,8 @@ A set consisting of a single point.
|
|||
|
||||
Let $S$ be a set consisting of a single point.
|
||||
By definition of a point, $S$ is a rectangle in which all vertices coincide.
|
||||
By \nameref{sec:choice-scale}, $S$ is measurable with area its width times
|
||||
its height.
|
||||
By \nameref{sec:area-choice-scale}, $S$ is measurable with area its width
|
||||
times its height.
|
||||
The width and height of $S$ is trivially zero.
|
||||
Therefore $a(S) = (0)(0) = 0$.
|
||||
|
||||
|
|
@ -698,7 +698,7 @@ A set consisting of a finite number of points in a plane.
|
|||
By construction, $S_{k+1} = S_k \cup T$.
|
||||
By the induction hypothesis, $S_k$ is measurable with area $0$.
|
||||
By \nameref{sub:exercise-1.7.1a}, $T$ is measurable with area $0$.
|
||||
By the \nameref{sec:additive-property}, $S_k \cup T$ is
|
||||
By the \nameref{sec:area-additive-property}, $S_k \cup T$ is
|
||||
measurable, $S_k \cap T$ is measurable, and
|
||||
\begin{align}
|
||||
a(S_{k+1})
|
||||
|
|
@ -746,8 +746,8 @@ The union of a finite collection of line segments in a plane.
|
|||
Consider a set $S$ consisting of a single line segment in a plane.
|
||||
By definition of a line segment, $S$ is a rectangle in which one side has
|
||||
dimension $0$.
|
||||
By \nameref{sec:choice-scale}, $S$ is measurable with area its width $w$
|
||||
times its height $h$.
|
||||
By \nameref{sec:area-choice-scale}, $S$ is measurable with area its width
|
||||
$w$ times its height $h$.
|
||||
Therefore $a(S) = wh = 0$.
|
||||
Thus $P(1)$ holds.
|
||||
|
||||
|
|
@ -761,7 +761,7 @@ The union of a finite collection of line segments in a plane.
|
|||
By construction, $S_{k+1} = S_k \cup T$.
|
||||
By the induction hypothesis, $S_k$ is measurable with area $0$.
|
||||
By the base case, $T$ is measurable with area $0$.
|
||||
By the \nameref{sec:additive-property}, $S_k \cup T$ is measurable,
|
||||
By the \nameref{sec:area-additive-property}, $S_k \cup T$ is measurable,
|
||||
$S_k \cap T$ is measurable, and
|
||||
\begin{align}
|
||||
a(S_{k+1})
|
||||
|
|
@ -823,10 +823,10 @@ Prove that every triangular region is measurable and that its area is one half
|
|||
\centering
|
||||
\end{figure}
|
||||
|
||||
By \nameref{sec:choice-scale}, both $R$ and $S$ are measurable.
|
||||
By \nameref{sec:area-choice-scale}, both $R$ and $S$ are measurable.
|
||||
By this same axiom, $a(R) = ab$ and $a(S) = ca\sin{\theta}$.
|
||||
By the \nameref{sec:additive-property}, $R \cup S$ and $R \cap S$ are both
|
||||
measurable.
|
||||
By the \nameref{sec:area-additive-property}, $R \cup S$ and $R \cap S$ are
|
||||
both measurable.
|
||||
$a(R \cap S) = a(T)$ and $a(R \cup S)$ can be determined by noting that
|
||||
$R$'s construction implies identity $a(R) = 2a(T)$.
|
||||
Therefore
|
||||
|
|
@ -839,8 +839,8 @@ Prove that every triangular region is measurable and that its area is one half
|
|||
& = ab + ca\sin{\theta} - ca\sin{\theta} - a(T).
|
||||
\end{align*}
|
||||
Solving for $a(T)$ gives the desired identity: $$a(T) = \frac{1}{2}ab.$$
|
||||
By \nameref{sec:invariance-under-congruence}, $a(T') = a(T)$, concluding our
|
||||
proof.
|
||||
By \nameref{sec:area-invariance-under-congruence}, $a(T') = a(T)$, concluding
|
||||
our proof.
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
|
@ -866,10 +866,10 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
|
|||
Suppose $S$ is a right trapezoid.
|
||||
Then $S$ is the union of non-overlapping rectangle $R$ of width $b_1$ and
|
||||
height $h$ with right triangle $T$ of base $b_2 - b_1$ and height $h$.
|
||||
By \nameref{sec:choice-scale}, $R$ is measurable.
|
||||
By \nameref{sec:area-choice-scale}, $R$ is measurable.
|
||||
By \nameref{sec:exercise-1.7.2}, $T$ is measurable.
|
||||
By the \nameref{sec:additive-property}, $R \cup T$ and $R \cap T$ are both
|
||||
measurable and
|
||||
By the \nameref{sec:area-additive-property}, $R \cup T$ and $R \cap T$ are
|
||||
both measurable and
|
||||
\begin{align*}
|
||||
a(S)
|
||||
& = a(R \cup T) \\
|
||||
|
|
@ -888,8 +888,8 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
|
|||
Then $R$ has longer base edge of length $b_2 - c$.
|
||||
By \nameref{sec:exercise-1.7.2}, $T$ is measurable.
|
||||
By Case 1, $R$ is measurable.
|
||||
By the \nameref{sec:additive-property}, $R \cup T$ and $R \cap T$ are both
|
||||
measurable and
|
||||
By the \nameref{sec:area-additive-property}, $R \cup T$ and $R \cap T$ are
|
||||
both measurable and
|
||||
\begin{align*}
|
||||
a(S)
|
||||
& = a(T) + a(R) - a(R \cap T) \\
|
||||
|
|
@ -906,7 +906,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
|
|||
Let $c$ denote the length of base $T$.
|
||||
Reflect $T$ vertically to form another right triangle, say $T'$.
|
||||
Then $T' \cup R$ is an acute trapezoid.
|
||||
By \nameref{sec:invariance-under-congruence},
|
||||
By \nameref{sec:area-invariance-under-congruence},
|
||||
\begin{equation}
|
||||
\label{sec:exercise-1.7.3-eq1}
|
||||
\tag{3.1}
|
||||
|
|
@ -933,7 +933,7 @@ Prove that every trapezoid and every parallelogram is measurable and derive the
|
|||
Let $c$ denote the length of base $T$.
|
||||
Reflect $T$ vertically to form another right triangle, say $T'$.
|
||||
Then $T' \cup R$ is an acute trapezoid.
|
||||
By \nameref{sec:invariance-under-congruence},
|
||||
By \nameref{sec:area-invariance-under-congruence},
|
||||
\begin{equation}
|
||||
\label{sec:exercise-1.7.3-eq2}
|
||||
\tag{3.2}
|
||||
|
|
@ -970,7 +970,7 @@ Prove that the formula is valid for rectangles with sides parallel to the
|
|||
We assume $P$ has three non-collinear points, ruling out any instances of
|
||||
points or line segments.
|
||||
|
||||
By \nameref{sec:choice-scale}, $P$ is measurable with area $a(P) = wh$.
|
||||
By \nameref{sec:area-choice-scale}, $P$ is measurable with area $a(P) = wh$.
|
||||
By construction, $P$ has $I = (w - 1)(h - 1)$ interior lattice points and
|
||||
$B = 2(w + h)$ lattice points on its boundary.
|
||||
The following shows the lattice point area formula is in agreement with
|
||||
|
|
@ -1082,7 +1082,7 @@ Use induction on the number of edges to construct a proof for general polygons.
|
|||
& = a(S) + (I_T + \frac{1}{2}B_T - 1) & \text{induction hypothesis} \\
|
||||
& = a(S) + a(T). & \text{base case}
|
||||
\end{align*}
|
||||
By the \nameref{sec:additive-property}, $S \cup T$ is measurable,
|
||||
By the \nameref{sec:area-additive-property}, $S \cup T$ is measurable,
|
||||
$S \cap T$ is measurable, and
|
||||
\begin{align*}
|
||||
a(P)
|
||||
|
|
@ -1681,4 +1681,227 @@ This property is described by saying that every step function is a linear
|
|||
|
||||
\end{proof}
|
||||
|
||||
\chapter{Properties of the Integral of a Step Function}%
|
||||
\label{chap:properties-integral-step-function}
|
||||
|
||||
\section{\partial{Additive Property}}%
|
||||
\label{sec:step-additive-property}
|
||||
|
||||
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval $[a, b]$.
|
||||
Then
|
||||
$$\int_a^b \left[ s(x) + t(x) \right] \mathop{dx} =
|
||||
\int_a^b s(x) \mathop{dx} + \int_a^b t(x) \mathop{dx}.$$
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||
$P_s$ such that $s$ is constant on each open subinterval of $P_s$.
|
||||
Likewise, there exists a partition $P_t$ such that $t$ is constant on each
|
||||
open subinterval of $P_t$.
|
||||
Therefore $s + t$ is a step function with step partition
|
||||
$$P = P_s \cup P_t = \{x_0, x_1, \ldots, x_n\},$$ the common refinement of
|
||||
$P_s$ and $P_t$ with subdivision points $x_0$, $x_1$, $\ldots$, $x_n$.
|
||||
|
||||
$s$ and $t$ remain constant on every open subinterval of $P$.
|
||||
Let $s_k$ denote the value of $s$ on the $k$th open subinterval of $P$.
|
||||
Let $t_k$ denote the value of $t$ on the $k$th open subinterval of $P$.
|
||||
By definition of the \nameref{sec:def-integral-step-function},
|
||||
\begin{align*}
|
||||
\int_a^b \left[ s(x) + t(x) \right] \mathop{dx}
|
||||
& = \sum_{k=1}^n (s_k + t_k) \cdot (x_k - x_{k-1}) \\
|
||||
& = \sum_{k=1}^n \left[ s_k \cdot (x_k - x_{k-1}) +
|
||||
t_k \cdot (x_k - x_{k-1}) \right] \\
|
||||
& = \sum_{k=1}^n s_k \cdot (x_k - x_{k-1}) +
|
||||
\sum_{k=1}^n t_k \cdot (x_k - x_{k-1}) \\
|
||||
& = \int_a^b s(x) \mathop{dx} + \int_a^b t(x) \mathop{dx}.
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{\partial{Homogeneous Property}}%
|
||||
\label{sec:step-homogeneous-property}
|
||||
|
||||
Let $s$ be a \nameref{sec:def-step-function} on closed interval $[a, b]$.
|
||||
For every real number $c$, we have
|
||||
$$\int_a^b c \cdot s(x) \mathop{dx} = c\int_a^b s(x) \mathop{dx}.$$
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ be a step function on closed interval $[a, b]$.
|
||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||
$P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open
|
||||
subinterval of $P$.
|
||||
Then $c \cdot s$ is a step function with step partition $P$.
|
||||
|
||||
By definition of the \nameref{sec:def-integral-step-function},
|
||||
\begin{align*}
|
||||
\int_a^b c \cdot s(x) \mathop{dx}
|
||||
& = \sum_{k=1}^n (c \cdot s(x)) \cdot (x_k - x_{k-1}) \\
|
||||
& = \sum_{k=1}^n c \cdot (s(x) \cdot (x_k - x_{k-1})) \\
|
||||
& = c \sum_{k=1}^n s(x) \cdot (x_k - x_{k-1}) \\
|
||||
& = c \int_a^b s(x) \mathop{dx}.
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{\partial{Linearity Property}}%
|
||||
\label{sec:step-linearity-property}
|
||||
|
||||
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval $[a, b]$.
|
||||
For every real $c_1$ and $c_2$, we have
|
||||
$$\int_a^b \left[ c_1s(x) + c_2t(x) \right] \mathop{dx} =
|
||||
c_1\int_a^b s(x) \mathop{dx} + c_2\int_a^b t(x) \mathop{dx}.$$
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
||||
Let $c_1$ and $c_2$ be real numbers.
|
||||
Then $c_1 \cdot s$ and $c_2 \cdot t$ are step functions.
|
||||
Then
|
||||
\begin{align*}
|
||||
& \int_a^b \left[ c_1s(x) + c_2t(x) \right] \mathop{dx} \\
|
||||
& = \int_a^b c_1s(x) \mathop{dx} + \int_a^b c_2t(x) \mathop{dx}
|
||||
& \textref{sec:step-additive-property} \\
|
||||
& = c_1\int_a^b s(x) \mathop{dx} + c_2\int_a^b t(x). \mathop{dx}
|
||||
& \textref{sec:step-homogeneous-property}
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{\partial{Comparison Theorem}}%
|
||||
\label{sec:step-comparison-theorem}
|
||||
|
||||
Let $s$ and $t$ be \nameref{sec:def-step-function}s on closed interval $[a, b]$.
|
||||
If $s(x) < t(x)$ for every $x$ in $[a, b]$, then
|
||||
$$\int_a^b s(x) \mathop{dx} < \int_a^b t(x) \mathop{dx}.$$
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ and $t$ be step functions on closed interval $[a, b]$.
|
||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||
$P_s$ such that $s$ is constant on each open subinterval of $P_s$.
|
||||
Likewise, there exists a partition $P_t$ such that $t$ is constant on each
|
||||
open subinterval of $P_t$.
|
||||
Let $$P = P_s \cup P_t = \{x_0, x_1, \ldots, x_n\}$$ be the common refinement
|
||||
of $P_s$ and $P_t$ with subdivision points $x_0$, $x_1$, $\ldots$, $x_n$.
|
||||
|
||||
By construction, $P$ is a step partition for both $s$ and $t$.
|
||||
Thus $s$ and $t$ remain constant on every open subinterval of $P$.
|
||||
Let $s_k$ denote the value of $s$ on the $k$th open subinterval of $P$.
|
||||
Let $t_k$ denote the value of $t$ on the $k$th open subinterval of $P$.
|
||||
By definition of the \nameref{sec:def-integral-step-function},
|
||||
\begin{align*}
|
||||
\int_a^b s(x) \mathop{dx}
|
||||
& = \sum_{k=1}^n s_k \cdot (x_k - x_{k-1}) \\
|
||||
& < \sum_{k=1}^n t_k \cdot (x_k - x_{k-1}) & \text{by hypothesis} \\
|
||||
& = \int_a^b t(x) \mathop{dx}.
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{\partial{Additivity With Respect to the Interval of Integration}}%
|
||||
\label{sec:step-additivity-with-respect-interval-integration}
|
||||
|
||||
Let $s$ be a \nameref{sec:def-step-function} on closed interval $[a, b]$.
|
||||
Then
|
||||
$$\int_a^c s(x) \mathop{dx} + \int_c^b s(x) \mathop{dx} =
|
||||
\int_a^b s(x) \mathop{dx} \quad\text{if}\quad a < c < b.$$
|
||||
|
||||
\todo{This holds for any arrangement of values $a$, $b$, and $c$.}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ be a step function on closed interval $[a, b]$.
|
||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||
$P$ such that $s$ is constant on each open subinterval of $P$.
|
||||
|
||||
Let $Q = \{x_0, x_1, \ldots, x_n\}$ be a refinement of $P$ that includes $c$
|
||||
as a subdivision point.
|
||||
Then $Q$ is a step partition of $s$ and there exists some $0 < i < n$ such
|
||||
that $x_i = c$.
|
||||
Let $s_k$ denote the value of $s$ on the $k$th open subinterval of $Q$.
|
||||
By definition of the \nameref{sec:def-integral-step-function},
|
||||
\begin{align*}
|
||||
\int_a^b s(x) \mathop{dx}
|
||||
& = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1}) \\
|
||||
& = \sum_{k=1}^i s_k \cdot (x_k - x_{k - 1}) +
|
||||
\sum_{k=i+1}^n s_k \cdot (x_k - x_{k - 1}) \\
|
||||
& = \int_a^c s(x) \mathop{dx} + \int_c^b s(x) \mathop{dx}.
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{\partial{Invariance Under Translation}}%
|
||||
\label{sec:step-invariance-under-translation}
|
||||
|
||||
Let $s$ be a step function on closed interval $[a, b]$.
|
||||
Then
|
||||
$$\int_a^b s(x) \mathop{dx} =
|
||||
\int_{a+c}^{b+x} s(x - c) \mathop{dx} \quad\text{for every real } c.$$
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ be a step function on closed interval $[a, b]$.
|
||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||
$P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open
|
||||
subinterval of $P$.
|
||||
Let $s_k$ denote the value of $s$ on the $k$th open subinterval of $P$.
|
||||
|
||||
Let $c$ be a real number.
|
||||
Then $t(x) = s(x - c)$ is a step function on closed interval $[a + c, b + c]$
|
||||
with partition $Q = \{x_0 + c, x_1 + c, \ldots, x_n + c\}$.
|
||||
Furthermore, $t$ is constant on each open subinterval of $Q$.
|
||||
Let $t_k$ denote the value of $t$ on the $k$th open subinterval of $Q$.
|
||||
By construction, $t_k = s_k$.
|
||||
|
||||
By definition of the \nameref{sec:def-integral-step-function},
|
||||
\begin{align*}
|
||||
\int_{a+c}^{b+c} s(x - c) \mathop{dx}
|
||||
& = \int_{a+c}^{b+c} t(x) \mathop{dx} \\
|
||||
& = \sum_{k=1}^n t_k \cdot ((x_k + c) - (x_{k - 1} + c)) \\
|
||||
& = \sum_{k=1}^n t_k \cdot (x_k - x_{k - 1}) \\
|
||||
& = \sum_{k=1}^n s_k \cdot (x_k - x_{k - 1}) \\
|
||||
& = \int_a^b s(x) \mathop{dx}.
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\section{\partial{Expansion or Contraction of the Interval of Integration}}%
|
||||
\label{sec:step-expansion-contraction-interval-integration}
|
||||
|
||||
Let $s$ be a step function on closed interval $[a, b]$.
|
||||
Then
|
||||
$$\int_{ka}^{kb} s \left( \frac{x}{k} \right) \mathop{dx} =
|
||||
k \int_a^b s(x) \mathop{dx} \quad\text{for every } k > 0.$$
|
||||
|
||||
\todo{This also holds for negative values of $k$.}
|
||||
|
||||
\begin{proof}
|
||||
|
||||
Let $s$ be a step function on closed interval $[a, b]$.
|
||||
By definition of a step function, there exists a \nameref{sec:def-partition}
|
||||
$P = \{x_0, x_1, \ldots, x_n\}$ such that $s$ is constant on each open
|
||||
subinterval of $P$.
|
||||
Let $s_i$ denote the value of $s$ on the $i$th open subinterval of $P$.
|
||||
|
||||
Let $k > 0$ be a real number.
|
||||
Then $t(x) = s(x / k)$ is a step function on closed interval $[ka, kb]$
|
||||
with partition $Q = \{kx_0, kx_1, \ldots, kx_n\}$.
|
||||
Furthermore, $t$ is constant on each open subinterval of $Q$.
|
||||
Let $t_i$ denote the value of $t$ on the $i$th open subinterval of $Q$.
|
||||
By construction, $t_i = s_i$.
|
||||
|
||||
By definition of the \nameref{sec:def-integral-step-function},
|
||||
\begin{align*}
|
||||
\int_{ka}^{kb} s(x / k) \mathop{dx}
|
||||
& = \int_{ka}^{kb} t(x) \mathop{dx} \\
|
||||
& = \sum_{i=1}^n t_i \cdot (kx_i - kx_{i-1}) \\
|
||||
& = k \sum_{i=1}^n t_i \cdot (x_i - x_{i-1}) \\
|
||||
& = k \sum_{i=1}^n s_i \cdot(x_i - x_{i-1}) \\
|
||||
& = k \int_a^b s(x) \mathop{dx}.
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
\end{document}
|
||||
|
|
|
|||
19
preamble.tex
19
preamble.tex
|
|
@ -23,30 +23,27 @@
|
|||
% Environments
|
||||
% ========================================
|
||||
|
||||
\newcommand{\divider}{\vspace{10pt}\hrule\vspace{10pt}}
|
||||
\newcommand{\header}[2]{\title{#1}\author{#2}\date{}\maketitle}
|
||||
|
||||
\newenvironment{axiom}{%
|
||||
\paragraph{\normalfont\normalsize\textit{Axiom.}}}
|
||||
{\hfill$\square$}
|
||||
\newenvironment{definition}{%
|
||||
\paragraph{\normalfont\normalsize\textit{Definition.}}}
|
||||
{\hfill$\square$}
|
||||
\newcommand{\divider}{%
|
||||
\vspace{10pt}
|
||||
\hrule
|
||||
\vspace{10pt}}
|
||||
\newcommand{\header}[2]{%
|
||||
\title{#1}
|
||||
\author{#2}
|
||||
\date{}
|
||||
\maketitle}
|
||||
\newcommand{\note}[1]{%
|
||||
|
||||
\newcommand{\admonition}[2]{%
|
||||
\begin{center}
|
||||
\doublebox{
|
||||
\begin{minipage}{0.95\textwidth}
|
||||
\vspace{2pt}
|
||||
\hl{Note:} #1
|
||||
\hl{#1} #2
|
||||
\vspace{2pt}
|
||||
\end{minipage}}
|
||||
\end{center}}
|
||||
\newcommand{\note}[1]{\admonition{Note:}{#1}}
|
||||
\newcommand{\todo}[1]{\admonition{TODO:}{#1}}
|
||||
|
||||
% ========================================
|
||||
% Status
|
||||
|
|
|
|||
Loading…
Reference in New Issue