daviddoji/CalculusWithJuliaNotes.jl
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

use quarto, not Pluto to render pages

Browse Source
This commit is contained in:
jverzani
2022-07-24 16:38:24 -04:00
parent 93c993206a
commit 7b37ca828c
879 changed files with 793311 additions and 2678 deletions
Download Patch File Download Diff File Expand all files Collapse all files

41
CwJ/integrals/improper_integrals.jmd
View File

@@ -17,7 +17,7 @@ const frontmatter = (
description = "Calculus with Julia: Improper Integrals",
tags = ["CalculusWithJulia", "integrals", "improper integrals"],
);
fig_size=(600, 400)
fig_size=(800, 600)
nothing
```
@@ -108,10 +108,8 @@ For the interval $(-\infty, \infty)$ we have need *both* these limits to exist,
\int_{-\infty}^\infty f(x) dx = \lim_{M \rightarrow -\infty} \int_M^a f(x) dx + \lim_{M \rightarrow \infty} \int_a^M f(x) dx.
```
```julia; echo=false
note("""When the integral exists, it is said to *converge*. If it doesn't exist, it is said to *diverge*.
""")
```
!!! note
When the integral exists, it is said to *converge*. If it doesn't exist, it is said to *diverge*.
##### Examples
@@ -201,13 +199,8 @@ Suppose $a < c$, we define $\int_a^c f(x) dx = \lim_{M \rightarrow c-} \int_a^c
= \lim_{M \rightarrow 0+} 2(1) - 2\sqrt{M} = 2.
```
```julia; echo=false
note(L"""
The cases $f(x) = x^{-n}$ for $n > 0$ are tricky to keep straight. For $n > 1$, the functions can be integrated over $[1,\infty)$, but not $(0,1]$. For $0 < n < 1$, the functions can be integrated over $(0,1]$ but not $[1, \infty)$.
""")
```
!!! note
The cases $f(x) = x^{-n}$ for $n > 0$ are tricky to keep straight. For $n > 1$, the functions can be integrated over $[1,\infty)$, but not $(0,1]$. For $0 < n < 1$, the functions can be integrated over $(0,1]$ but not $[1, \infty)$.
* Now consider $f(x) = 1/x$. Is this integral $\int_0^1 1/x \cdot dx$ defined? It will be *if* this limit exists:
@@ -417,8 +410,8 @@ choices =[
"It is convergent",
"It is divergent",
"Can't say"]
ans = 1
radioq(choices, ans, keep_order=true)
answ = 1
radioq(choices, answ, keep_order=true)
```
@@ -433,8 +426,8 @@ choices =[
"It is convergent",
"It is divergent",
"Can't say"]
ans = 3
radioq(choices, ans, keep_order=true)
answ = 3
radioq(choices, answ, keep_order=true)
```
----
@@ -450,8 +443,8 @@ choices =[
"It is convergent",
"It is divergent",
"Can't say"]
ans = 2
radioq(choices, ans, keep_order=true)
answ = 2
radioq(choices, answ, keep_order=true)
```
----
@@ -465,8 +458,8 @@ choices =[
"It is convergent",
"It is divergent",
"Can't say"]
ans = 2
radioq(choices, ans, keep_order=true)
answ = 2
radioq(choices, answ, keep_order=true)
```
----
@@ -480,8 +473,8 @@ choices =[
"It is convergent",
"It is divergent",
"Can't say"]
ans = 1
radioq(choices, ans, keep_order=true)
answ = 1
radioq(choices, answ, keep_order=true)
```
###### Question
@@ -496,8 +489,8 @@ choices = [
"``\\int_0^1 u^{2/3} \\cdot du``",
"``\\int_0^\\infty 1/u \\cdot du``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question