daviddoji/CalculusWithJuliaNotes.jl
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use quarto, not Pluto to render pages

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jverzani
2022-07-24 16:38:24 -04:00
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CwJ/integrals/volumes_slice.jmd
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@@ -105,13 +105,8 @@ area is $\pi r(x)^2$ so the volume is given by:
V = \int_a^b \pi r(x)^2 dx.
```
```julia; echo=false
note(L"""
The formula is for a rotation around the $x$-axis, but can easily be generalized to rotating around any line (say the $y$-axis or $y=x$, ...) just by adjusting what $r(x)$ is taken to be.
""")
```
!!! note
The formula is for a rotation around the $x$-axis, but can easily be generalized to rotating around any line (say the $y$-axis or $y=x$, ...) just by adjusting what $r(x)$ is taken to be.
For a numeric example, we consider the original Red
@@ -136,8 +131,8 @@ This is
```julia;
d0, d1, h = 2.5, 3.75, 4.75
r(x) = d0/2 + (d1/2 - d0/2)/h * x
vol, _ = quadgk(x -> pi * r(x)^2, 0, h)
rad(x) = d0/2 + (d1/2 - d0/2)/h * x
vol, _ = quadgk(x -> pi * rad(x)^2, 0, h)
```
So $36.9 \text{in}^3$. How many ounces is that? It is useful to know
@@ -178,7 +173,7 @@ So we need to solve $v \cdot (231/128) = \int_0^h\pi r(x)^2 dx$ for $h$ when $v=
Let's express volume as a function of $h$:
```julia;
Vol(h) = quadgk(x -> pi * r(x)^2, 0, h)[1]
Vol(h) = quadgk(x -> pi * rad(x)^2, 0, h)[1]
```
Then to solve we have:
@@ -201,12 +196,8 @@ As a percentage of the total height, these are:
h5/h, h12/h
```
```julia;echo=false
note("""
Were performance at issue, Newton's method might also have been considered here, as the derivative is easily computed by the fundamental theorem of calculus.
""")
```
!!! note
Were performance at issue, Newton's method might also have been considered here, as the derivative is easily computed by the fundamental theorem of calculus.
##### Example
@@ -239,16 +230,11 @@ quadgk(x -> pi*radius(x)^2, 1, Inf)[1]
That is a value very reminiscent of $\pi$, which it is as $\int_1^\infty 1/x^2 dx = -1/x\big|_1^\infty=1$.
```julia; echo=false
note("""
The interest in this figure is that soon we will be able to show that
it has **infinite** surface area, leading to the
[paradox](http://tinyurl.com/osawwqm) that it seems possible to fill
it with paint, but not paint the outside.
""")
```
!!! note
The interest in this figure is that soon we will be able to show that
it has **infinite** surface area, leading to the
[paradox](http://tinyurl.com/osawwqm) that it seems possible to fill
it with paint, but not paint the outside.
##### Example
@@ -668,8 +654,8 @@ choices = [
"``1/3 \\cdot w^2\\cdot h``",
"``l\\cdot w \\cdot h/ 3``"
]
ans = 2
radioq(choices, ans)
answ = 2
radioq(choices, answ)
```
###### Question
@@ -696,8 +682,8 @@ choices = [
"``4/3 \\cdot \\pi a^2 b``",
"``\\pi/3 \\cdot a b^2``"
]
ans = 1
radioq(choices, ans)
answ = 1
radioq(choices, answ)
```
###### Question
@@ -752,9 +738,9 @@ Find the volume of rotating the region bounded by the line $y=x$, $x=1$ and the
```julia; hold=true; echo=false
cm=[2/3, 1/3]
c = [1/2, 1/2]
r = norm(cm - c)
rr = norm(cm - c)
A = 1/2 * 1 * 1
val = 2pi*r*A
val = 2pi * rr * A
numericq(val)
```