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pluto_riemann.jl
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pluto_riemann.jl
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### A Pluto.jl notebook ###
# v0.19.36
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 41117df0-f9f2-11ec-0eaf-bbe82f267d42
begin
using Plots
using PlutoUI
using LinearAlgebra
using Symbolics
using LaTeXStrings
end
# ╔═╡ dbed234b-ebf7-4448-889a-8c98cacbeda9
md"""
### Riemann Integrability
Let $f:[a,b] \rightarrow [m,M]$ be a real-valued bounded function on an interval $[a,b]$, that is, the graph of $f$ is contained in a closed rectangle $[a,b]\times[m,M]$.
"""
# ╔═╡ 0e86eec9-4dec-4837-8b5f-8fe86cd688b9
begin
x = range(0, 1, length = 101);
f=sin.(2*pi*x)+x+1.2*ones(101);
z=1.8*ones(101);
w=zeros(101);
for k=1:101
w[k]=min(f[k],z[k])
end
end
# ╔═╡ 03147458-df8f-4526-bc60-715071c3ac16
begin
plot(f,
title=L"y=f(x)",
lw=2,
grid=false,
xlim=[-10,111],
ylim=[0.4,2.8],
yticks=([0.6 2.7;],[L"m",L"M"]),
xticks=([1 101;],[L"a",L"b"]),
xlabel=(L"x"),
ylabel=(L"y"),
legend=false)
plot!([1,1],[0.6,2.7], c=:magenta)
plot!([1,101],[2.7,2.7], c=:magenta)
plot!([101,101],[2.7,0.6], c=:magenta)
plot!([101,1],[0.6,0.6], c=:magenta)
end
# ╔═╡ f13630b1-420d-40d4-b24f-c100d29fb30e
md"""
We define a Riemann sum of $f$. We introduce division of $[a,b]$
$\Delta: a=a_0<a_1<\dotsb<a_n=b, \quad \lvert{\Delta}\rvert:=\max_{i=1,\dotsc,n}(a_i-a_{i-1}).$
Pick up arbitrary $\xi_i\in[a_{i-1},a_i]$ for $i=1,\dotsc,n$. Set
$R(\Delta):=\sum_{i=1}^nf(\xi_i)(a_i-a_{i-1}),$
This is a Riemann sum of $f$. This depends not only on the division $\Delta$ but also the choice of $\xi_1,\dotsc,\xi_n$. Note that
$m(b-a) \leqq R(\Delta) \leqq M(b-a).$
We say that $f$ is Riemann integrable if there exists a cosntant $\alpha$ such that
$R(\Delta) \rightarrow \alpha \quad (\lvert\Delta\rvert \rightarrow 0),$
which is independent of the choices of $\Delta$ and $\xi_1,\dotsc,\xi_n$.
In this case the real number $\alpha$ is said to be the Riemann integral of $f$ on the interval $[a,b]$ and is denoted by
$\int_a^bf(x)dx:=\alpha.$
It is wellknown that if $f \in C[a,b]$, then $f$ is uniformly continuous on $[a,b]$ and Riemann integrable on $[a,b]$. More generally if there exist finite numbers
$b_0,b_1,\dotsc,b_n\in[a,b]$ such that $a=b_0<b_1<\dotsb<b_N=b$, $f \in C(b_{j-1},b_j)$,
$j=1,\dotsc,N$ and limits of finite values
$f(b_0+0), \quad f(b_1-0), \quad f(b_1+0), \quad \dotsc \quad f(b_N-0)$
exit, that is, $f$ is piecewise continuous on $[a,b]$, then $f$ is Riemann integrable on $[a,b]$.
"""
# ╔═╡ 174c82bf-34bd-47e7-bb52-93307f29b47c
md"""
n= $(@bind n Slider(1:300, show_value=true))
"""
# ╔═╡ 24d518cb-7088-4efb-a87b-6882fb17c5f9
begin
x1 = range(0, 1, length = 501);
f1=sin.(2*pi*x1)+x1+1.2*ones(501);
g1=zeros(501);
for j=1:501
g1[j]=f1[Int64(floor(floor(n*j/501)*501/2/n)+ceil(ceil(n*j/501)*501/2/n))];
end
end
# ╔═╡ 7b34916d-8ab9-4157-a5fd-9f45fb2915ef
begin
plot(f1,
grid=false,
ylim=(0,2.7),
xticks=([1 501;],[L"a",L"b"]),
yticks=([0;],[0]),
xlabel=(L"x"),
ylabel=(L"y"),
edgecolor=:false,
legend=false,
lw=3,
color=:blue)
bar!(g1,lw=0,color=:skyblue)
end
# ╔═╡ c7e78791-8e99-4615-8f10-13fab72d03e1
md"""
If $f(x)$ is continuous and nonnegative, then the Riemann integral $\displaystyle\int_a^bf(x)dx$ is the area unclosed by $y=0$, $x=a$, $x=b$, and $y=f(x)$.
"""
# ╔═╡ 7c32e837-aaf4-402c-b4b2-1985441b23ea
begin
plot(f,
lw=2,
grid=false,
fillrange=0,
fillcolor=:lightgoldenrod,
ylim=[0,2.7],
yticks=([0;],[0]),
xticks=([0 101;],[L"a",L"b"]),
xlabel=(L"x"),
ylabel=(L"y"),
legend=false)
plot!([1,1],[0,f[1]], c=:magenta)
plot!([101,101],[0,f[101]], c=:magenta)
plot!([1,101],[0,0], c=:red)
end
# ╔═╡ 0dec1d8a-79a1-4e1d-9df6-b5d39a10f9cd
md"""
Let $c \in (a,b)$. Suppose that $f(x)$ is nonnegative and continuous for $x{\ne}c$, and has a jump discontinuity at $x=c$. The Riemann integral $\displaystyle\int_a^bf(x)dx$ is the area unclosed by $y=0$, $x=a$, $x=b$, $y=f(x)$ and the line segment between two points $\bigl(c,f(c-0)\bigr)$ and $\bigl(c,f(c+0)\bigr)$.
"""
# ╔═╡ 6cb50fc5-b02b-4938-b6f5-73ebe0af9481
begin
h=sin.(2*pi*x)+x+1.2*ones(101);
for k=51:101
h[k]=f[k]-1.5*(1-x[k])
end
end
# ╔═╡ a280745e-f3e0-4ef0-a76b-1054eb695b64
begin
plot(h,
lw=2,
grid=false,
fillrange=0,
fillcolor=:lightgoldenrod,
ylim=[0,2.7],
yticks=([0;],[0]),
xticks=([0 51 101;],[L"a",L"c",L"b"]),
xlabel=(L"x"),
ylabel=(L"y"),
legend=false)
plot!([1,1],[0,f[1]], c=:magenta)
plot!([101,101],[0,f[101]], c=:magenta)
plot!([1,101],[0,0], c=:red)
plot!([50,51],[h[50],h[51]], c=:lightgreen,lw=3)
end
# ╔═╡ 85f480d6-4906-4983-95ae-df60e30bc563
md"""
If $f(x)$ is continuous and takes both positive and negative values, then
$\displaystyle\int_a^bf(x)dx
= \text{the area of positive part} - \text{the area of negative part}.$
"""
# ╔═╡ e34cd84b-07b1-4fb2-a18c-2545821a6530
begin
plot(f,
lw=2,
grid=false,
fillrange=0,
fillcolor=:lightgoldenrod,
ylim=[0.9,2.5],
yticks=([1.8;],[0]),
xticks=([0 101;],[L"a",L"b"]),
xlabel=(L"x"),
ylabel=(L"y"),
legend=:false)
plot!(z,
lw=2,
grid=false,
fillrange=0,
fillcolor=:lightpink)
plot!(w,
lw=2,
grid=false,
fillrange=0,
fillcolor=:white)
plot!([1,1],[1.8,f[1]], c=:magenta)
plot!([101,101],[1.8,f[101]], c=:magenta)
plot!([1,101],[1.8,1.8], c=:red,lw=2)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
[compat]
LaTeXStrings = "~1.3.1"
Plots = "~1.38.17"
PlutoUI = "~0.7.52"
Symbolics = "~5.5.1"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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