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in this episode you’ll learn how to use python to find the volume of a solid

using the washer method hi everyone what’s up let’s do some math [Music]

hi everyone welcome back we’re going to begin by talking uh about what the

washer method is um and um you know before we start

though i wanted to mention that this episode is part of the series

applications of integration complete in-depth tutorials for calculus

so in a previous episode we talked about the washer method

and this time in this episode i want to emphasize python programming language

but before we do that i want to just briefly recall

or you know talk about what we talked about in the last episode so we’re going

to say a of x is the cross-sectional area of a disk and we’re going to have

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an outer radius and an inter radius so let me sketch the graph and explain to

you what i’m uh what i mean by this right here so let’s say we have a or

axis right here so let’s say here we got this and this and let’s say we got a

function coming through here let’s use blue for the upper function right here

so let’s say this is f of x here and then we’re going to have a lower curve

here g of x coming through here so that would be g of x right there

and then we’re going to be talking about a point right here and so we got a

boundary right here x equals a and x equals b

and what we’re going to do is we’re going to revolve this around the x-axis and

so we’re looking at this region right here which will shade in orange right here

just real quick and now you know when we revolve this region around the

x-axis what we’re going to have is a volume we’re going to have a solid we’re

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going to find that volume of that solid so

when we pick an x value right here let’s just say we pick an x between a and b

and so what we’re what we’re looking at right here is this distance right here

um and so you know we have y equals f of x right here

and y equals g of x right here and so we got these two two heights right here

and we’re looking for this distance right in here

so we’re going to call this r of x right here the distance from the x-axis here

this is called the outer radius right here and this distance right here is

called the inner radius right here and so we want to find that um

you know we want to you know if we think about this right here we can go to a b

pi and then this will be r of x squared and this is uh what we did in the last

episode where we talked about the disk method and this will be dx

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and then we can integrate and you know have the inner radius so we’ll have minus

the integral from a to b pi and now we have the inner radius right here

and so that’ll be squared and so this is similar to

um in that episode where we talked about finding the area between two curves

we had the difference of two functions but now we’re having the difference of

two cross-sectional areas and so basically what we’re doing is

we’re using the disk method to revolve and get the full solid and then we’re

going to also do the the inner one right here and then we’re going to take away

those solids so think about it is this is the solid of the

full object and then we’re going to take away the the solid on the interior here

like we hollowed it out and so that right there is the idea behind why

we have a difference of two squares right here

so this is the cross sectional area here

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which goes all the way from here to here

then and then we have a a representative rectangle and that representative

rectangle is revolving and that gives us a disk when you

revolve it around the x-axis so we get a disk right here

and the radius right here is is the r of x right there

and then we’ll do the same thing right here we have a representative

and then we have a disk coming in here and we have a and this height is r of x

and this full height right here is capital r of x so that’s the capital

r of x so we have the inner radius and the outer radius so this distance right

here in red is r of x minus um yeah so anyways um

so when we revolve this let’s see if we can sketch a diagram of that

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let’s try to put it right i’ll just draw a little little doodle over here

and we got this coming in here f and we got the g coming in here and we got this

f coming in here it’s supposed to be symmetric and we got this coming in here

and this coming in here and then this right here is coming in here like this

and then we got this little part right in here and this part right in here

and so we’re removing this inner um solid right here where we’re moving

that out and that’s why this minus comes in here because when we revolve this

around here uh instead of having disks what we’re

going to be using is the washers so we’re going to have this right here

the capital r of x and then and then we’re going to have the little r of x

here’s inner and outer radiuses okay so i want to kind of help you understand

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that all right and so now let’s look at our first example right here

now when we look at these examples right here

i want to emphasize in this episode using the python method

because in the previous episode we did this by hand here and so you know

we can look at that i want to set up the python first here so let’s go to a

python notebook now if you’ve never used python before

then what i want to say is that there’s a link below in the description and you

can open up your first python notebook just by following that link in the

description now once you open up your python notebook

then what you’ll need to do is to click in a cell and type this up right in here

these are the packages that i’m going to use in this episode so we’ve used all

these packages right here before in previous episodes uh we’re using the

numpy package and the plot package because we’re going to we’re going to

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sketch the region that we’re trying to revolve and we’re going to use the sci

pi and we’re going to do some integration and as i previously said

previously said in all the other episodes hey where i’m going to

customize my axes to make this look nice so let’s execute those two

cells right there after you type in that

right there exactly as it appears on the screen

all right so then now we’re going to go down here to the washer method and

here’s going to be our first example use the washer method to find the volume of

the solid generated by this line and this curve right here

and so yeah we did this one right here in a previous episode so i want to focus

on the python right here so what i’m going to do is i’m going to

say this is 2 times square root of x and that’s going to be my function f so i’m

defining a function f and i’m going to use the numpy package

so i’m going to do mp dot square root so it’ll be 2 times the square root of x

and then for the other function it’ll just be the x the identity function here

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and so i’m going to call that g so g of x just returns an x

and then now we’re going to make a figure we’re going to

make some axes i’m going to customize my axes

then i’m going to create a bunch of x values

and i’m going to be looking on 0 to 4 so i’ll explain y 0 to 4 and i’m going to

make a thousand of them and for each of these x’s i’m going to plug it into the

function right here the square root function i’m going to plot that

and then i’m going to make some y’s from the x function

from the g function and then i’m going to plot that

and then now we’re going to fill between the x the y 1 and the y 2 to get our

region and i’m going to color it a certain way

here put an alpha channel on it so you can see what i’m pointing at here and um

yeah so then after that we’re going to show the plot

and then we’re going to actually do the calculus part right here

so here’s a sketch of the region right here square root of uh two square roots

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of x and here’s the line uh y equals x and we’re shading in between here here’s

our region and now why four four well because that’s where our intersection

points are so where are these two functions equal to each other right

so when x is four um you know you you can see that that

works right there you get uh when x is four you’re going to get 4 equals 4. so

let’s see here we’re going to integrate we’re going to use psi pi and we’re

going to integrate and i’m going to use the quad method

here now if you haven’t seen this lambda before all that means is that is an

anonymous function so here is a function called f here’s a function called g

and we could to go define this function down here but i’m just going to type it

up anonymously so i use the word lambda and my variable is going to be an x and

so here’s the function right here and so this is just pi times f of x squared

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minus the g of x squared and then i’m going to be working on zero to four and

i’m going to be interested in the in the numeric value for that right

there so i’m going to execute that cell right there and we see we get out 33.5

right there and so there is the um value of the

volume right there when you revolve this

around the x-axis right here so we’re in this problem right here we’re revolving

around about the x-axis right there so to set this up and solve this by hand

we would do you know volume equals and then this would be 0 to 4

and then we would have 2 square roots of x and we would have that squared

and so i forgot my pi so pi and then uh two square roots of x and that squared

and then we would have minus and then a pi and then we would have an x and that

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would be squared and then we would have dx so this is according to the um

you know washer method here where we’re going to use washers right here okay so

what we’re going to do here is just simply integrate that

and when you do we’re going to get pi and we’re going to get

2 x squared minus x to the third over 3 and then we’re going to go from 0 to 4

and then when we finish this out we’re going to get 32 pi over 3

and this is going to be the exact value right here

so this is the exact value of the volume that we’re going to get

when this right here this object right here is revolved around the x-axis so i’m

going to be revolving around the x-axis right there

and when we do that we’re going to get an object it’s going to be closed down

here and it’s just going to be kind of opened up here and it’s going to have

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some interior almost like a curved shaped cup or something like that

all right so there’s the difference between the

by hand you know you get the exact value and the numeric value now if you didn’t

use a 0 right here then what you get is a an array

and you’re going to get the first entry is the numeric value which we just had

and then we’re going to get the um you know this is really small this is this is

exponential scientific notation right and so this has got a lot of zeros on it

this is really close to zero and so it’s telling you that this is very accurate

right here all right so let’s look at a second example here

um so let’s go here and look at that right here so on this one right here

we’re going to use the washer method to find the volume of the solid generated

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so we have y equals x right here uh oh sorry that’s the previous one

sorry example 2. so we’re going to look at y equals 2 sine x

and the x axis and we’re going to be working on 0 to pi and we’re going to be

rotating around the line y equals -2 so in the previous example we did the

x-axis which was y equals 0 and so now we’re going to do

a different horizontal axis uh just to see how this works right here

so um let’s do this out by hand here if we can first before we go to python um

and so i’ll just you know go over pretty quickly because we did this in the

previous episode so just want to refresh

your memory how to do that but we have a sine function going right through here

um so this will be y equals 2 sine x and here the height is you know a 2 right

here and then this is the line right here so this is the x axis

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and this is the line right here y equals what minus 2

and we’re going to be revolving around this axis right here this axis right

here so we’re going to be taking this region right here let me shade it in

orange really quick we’re going to be taking this region right here and

revolving around the line y equals minus 2 here and so when we do that

we’re going to get a object that’s been hollowed out

like someone drilled through it or something and so you know this right here is

the origin right here and this is the height of two [Music]

and so you can kind of get a feeling for what this shape right here is it’s just

sine x but it’s been scaled a little bit

in any case we’re going to be looking on 0 to pi so 0 to pi

and we’re going to revolve this and so what happens when it gets revolved so

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this is 0 to 2 and then this is -2 right here um so

you know this is going to get revolved around here so this is going to

look something like this right here we’re going to need to make this be a 4

right and so we’re going to come down here

this distance is going to be a 2 right here so i’m going to need a 2 let’s put

a 2 first and then we’ll put the 4. so let’s put a 2 here and then we’ll put a 4

and so now i’m going to have the shape right here but upside down let’s see if

i can do that it’s just going to kind of go like that

and then this will be the line uh sorry this would be the line here

so this distance is 2 just like this distance is 2. so this will be minus four

and then this will be minus six right here

and so this right here it’s right here at the zero and the pi right there and

so when we get when this revolves around y equals minus two we’re getting this

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object the solid that’s been hollowed out here in the middle right there

so just so that you can kind of kind of visualize it there this will be a minus

4 and that will be a minus 6 right there all right so now we can visualize it a

little better we want to identify the outer radius and the inner radius so the

outer radius is going to be given by 2 plus 2 sine x

and the inner radius is just going to be given by a2 right so the outer radius

is um the distance from here to here let’s go ahead and put it in red right

so we’ll pick an x right here and we’re looking at this part right here right

and so we want to say what the r of x here so the r of x

is this distance all right here so it’s going to be this distance right here 2

and then come up to the height of the function which is 2 sine x so that’s why

that’s the outer radius there now the inner radius is just this

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distance right here so i’ll call this right here if i can make this symbol here

it’s not very good um there that’s close

enough r of x is that distance there how about i just use arrows yeah so this

distance right here is r of x that’s the inner radius

and this distance right here is the capital

that’s the capital of the outer radius right there

all right so let’s find the volume using

the washer method so we’re going to have a

i’m going to say 0 to pi and then we’re going to have a pi and i’ll i’ll say pi

i’ll do brackets and then pi and then we have our

r of x squared so 2 plus 2 sine x so all that squared

minus and then pi and then the inner radius which is just 2. so this will be

a 2 and then a squared and then all of that d of x so

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it’s not so bad when you’re first starting out and doing it like that that

you write it in a form that which you can make sense of it um

so here i have all of this this is everything that i’m integrating and so

this is pi r squared minus pi r squared of course if you’ve done you know 50 of

these you can start factoring out the pi on your first step or whatever but

anyways we’re gonna have a pi uh zero to pi and then we need to expand this out

right here and then we’re going to have a minus 4 pi right there

and so we’re going to or let’s say a 2 squared a 4 and then a minus 4

right in any case when you factor that out and simplify it we get 8 sine x

plus 4 sine squared x and after you do the

complete work you’re going to get 2 pi times pi plus eight

so we filled in the details in the previous episode i recommend checking

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that out but any case there’s the exact value right here

and now what i want to show you is you know how to use python

to get a numeric approximation for that um and to get a plot of the region there

and so yeah let’s go back to python let’s go back to our notebook here

and let’s get rid of all this but you know over here

let’s go ahead and put the exact value so we can just remember what that is

right there 2 pi times pi plus 8. and so yeah let me get rid of

this real quick and we’ll look at the python right so here we go so um we’re

looking at two sine x x axis zero to pi and we’re revolving around minus two

here all right so i’m going to make the function two

uh times sine x so that’s our function right there

and i’m going to make a plot some axes i’m going to customize my axes and then

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i’m going to make up a bunch of x values between 0 and pi

and i’m going to make up a thousand of them i’m going to plug each one of those

into f and then i’m going to plot it and then i’m going to fill between x and y

and there’s my plot right there all right so

there’s my plot right there and it’s just the sine x plot from zero to pi um

and so there’s the region right there now this plot right here doesn’t show

you all the details that we had on the on the when i did it by hand so um

[Music] you know doing this by hand is is a lot perhaps better in this example

but i mean you can we can tack that on here and sketch it out but we already

did that let’s just get on to the calculus part so we’re going to use psi

pi we’re going to integrate we’re going to use the quad method we’re going to do

an anonymous function here what’s the formula for finding the

for the washer method right for finding the volume so the formula for the washer

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method is uh pi times and then we’re going to have 2 plus and

the two plus comes from the fact that we’re revolving around the line y equals

minus two so we’re with the our axis of revolution has been shifted

so we have two plus f of x and then all that squared and then minus

and then here we have the um [Music] the why do we have minus two here um

yeah so minus two or two either way get the same numeric value but

so minus and then we’re going to use a 2 for that distance right there and then

squared and then we’re going to integrate from 0 to pi

and then i’m just going to be looking at the numeric value right there

all right and so there we go and so there’s where we get the

70.005 or something like that pretty close to 70 there

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all right so there’s uh example two um now let’s look at an example three

and let’s go back over here and look at this one right here

so we’re gonna use the washer method again and we’re gonna find the volume

bounded by the graph two sine oh no that’s the same one again my bad all

right so now let’s look at a vertical axis

so this time we’re going to be revolving around a vertical axis so instead of

revolving around horizontal now i’m going to revolve around a vertical axis so

what does that look like here so now i’ll try to sketch the graph

or show you what that would look like over here

and so now let me draw this one pretty straight here

and we’re looking right here and so now i’m going to say i got a function

coming in here now when i say function i don’t mean

a function of x i mean a function of y so it could come in something like this

right here and then come out and this one right here just kind of come in here

like that and sweep through there and let’s call this one here a function

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of y so let’s call it x equals w of y and let’s call this one right here

x equals let’s just call it v of y here and now we’ve got bounds along the y

axis so this right here will be a bound and you know what let’s just bound it up

and just just look at this part right here we don’t need to sketch it all the

way down there and so this this we’re going to say is a c

and this is going to be a d and so that’s going to give us our region right here

that we’re going to be bounded in between and let me shade this in orange right

here as i’ve been doing so far so going to be taking this region right here

that’s defined using these horizontal lines and these functions of y

and we’re going to revolve this around a horizontal axis now sorry a vertical

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axis now and we’re going to choose to do this around the y axis so we can see

that way first all right so there we go and when we uh revolve this around the y

axis now we’re going to get a solid we’re

going to get a solid and we’re going to revolve this around there so what would

the solid kind of look like so let’s call this here x equals

this one right here is just x equals v of y for this curve right here

in any case um yeah so here’s how we do the washer

method when we’re going to be using functions of y

so we’re going to be looking at the cross sectional area which will be r y

uh for the outer radius and our roi here for the um

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yeah so this right here should be um a lower case r and so yeah all right

so i was just fixing that right there in any case yeah so here’s going to be the

r of y right here that distance right there that’s the ry

and this distance right here is the um little r of y the inner radius right

there all right so i like that all right so this is how we’re going to

set it up and what we’re going to end up with is this um

solid um it looks something like this right it comes in here like that so it

comes in here like that and then it has this one right here like this one right

here and then um it has the outer one here let’s maybe

make it a little bit more curvier like that and then like like this right here

something like that and then and so at the bottom here it’s going to have a

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it’s going to have a bottom because it’s chopped off at the bottom right there

and it’s chopped off at the top right there and so let’s make this rounded right

here and looks like a vase except for the fact that

we got some separation right here so what’s going to happen is we’re going to

have a um we can draw a smaller one right here

and that’s the sides of this is going to be given by this function right here v

of y and so we can just say it comes in like that and then it comes in like that

and so we’ll have some smaller thing there and so what we’re looking at here is

this think of it as like a vase or something but then it’s been hollowed out

so it’s really solid but then it has this uh region in here right there

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all right so um yeah so here we go let’s look at an example here we go

so find the volume of the solid that results in the region enclosed by the

curves x equals 1 minus y squared x equals 2

plus y squared y equals 1 and y equals minus 1. so let’s look at that region

right there really quick if i could sketch a graph by hand and

so we have this um one minus y squared so that’s going to

come in here and look like this and so it’s like a parabola but sideways

and it’s got a minus right here so it’s opening up to the left and this right

here is going to be opening up to the right and so it’s going to be looking like

you know something similar but and so we’re going to have a

horizontal line right here at y equals 1 and a horizontal line right here at

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y equals minus 1. and so what we’re looking at here is this region right here

we’re going to shade this in right here and we’re we’re going to take this

region right here and revolve it around the y-axis right there and so this is

going to use the washer method not the disk method because this is not flush

with the axis of rotation so this is the y-axis right here and we’re

going to revolve it around the y axis right here and so how do we do that right

so we need to pick a y and think about what the r of y is right here

what is that distance right there in red and what is this distance right here so

this distance is little r of y and the big distance right here is capital r y

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and so what we’re going to do is we’re going to say r of y is 2 plus y squared

right so we got a function of y right here and our lowercase of y is the

1 minus y squared and so we’re going to use the washer

method here to find the volume so the volume will be pi

and we’re going to integrate here from what to what um so

a little sloppy right here wasn’t i let’s make that a little bit better

right there this is going to come down here and meet us right here there we go

and so we’re going to be integrating here this is a height of minus 1 right

there y equals minus 1 and this is a height of 1 right here so we’re going to

be integrating from minus 1 to 1. and all right so i pulled out my pi

already so i need my outer radius 2 plus y squared and then

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we need to square that and then i have 1 minus y squared and we

need to square that and then all of that right there so

close off the bracket and then we have d y right there

and so there’s how we would set that up um and then one intermediate step for

you i’ll put here will be three plus six y squared d y and then

show some steps and then we get 10 pi so we already did this on a previous

episode so we got 10 pi when we’re finished so let’s look at uh how to do this

using python now so pause that right there if you need to

look at that anymore but anyways let’s look at the python there so here we go so

oh yeah let me get rid of this really quick so example three

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and we’re going to be looking at the python right here

all right so here we go we got the x and the x and the y equals 1 and the y

equals minus 1. it’s revolved around the y axis

now to do this i needed to break the up into pieces because

if we tried to solve this for y you know i mean if you just look at this

right here it’s just a it’s not a function right it’s not a function of x

but we could break it up into two functions of x so we could break up this

piece right here as a function of x which i’m calling f up

and we can look at this function of x right here just this lower branch right

here and we can call that a function of x which i called uh f down

and so that’ll be positive the square root so i’m just moving the y

over and then moving the x over and then taking the square root of that so in

other words just solving this for y right so it’s just like

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y squared equals one minus x and then taking square root

and you get plus or minus and so one for the plus one for the minus

all right and then we’ll do the same thing for the g for this one right here

we’ll do the same thing we’ll move the y squared over here we’ll move the x over

there multiply through by a negative in any case we get the g up and the g

down because this function right here is a parabola that opens up this way

and so we got the g up and the g down function right there

so there’s the four functions that we’re going to use

and for each one of these we’re going to um well before we do each one we’re

going to we’re going to define a figure axes customize axes and set the aspect

ratio so for each one of these the f up the f

down the g up and the g down functions i’m going to make a bunch of x’s

i’m going to plot the x’s and the y’s and then i’m going to fill it and i’m

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going to do that for each of them one two three four

so once we do all four of those then i’m going to fill in between

so those four does this part this part this part in this part and then the last

part is i need to fill in between one and two

and i need to go to a height of one and minus one to fill that in there oops

and so that will give us the whole region right there in yellow that we’re

going to revolve around the y-axis so there’s how i shaded that in

with python and plotted that function right here here’s f up here’s f down and

the same for g over here okay so now let’s integrate this

so we’re going to integrate from minus one to one there’s our bounds right there

and what are we going to be integrating so we’re going to be integrating a

function of y and so here’s the function right here is

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this is the outer right here so it’s 2 plus and then the y squared right here

so that’s the um r of y and then minus and then this is

the lower the lower radius the inner radius um and so this was 1 minus y squared

and then we’re going to square both of those and multiply it by pi

so we’re using the cross-sectional area right there

all right and so then yeah let’s go ahead and execute this cell right here

and then that’s where we get the uh 10 pi or 31.41 so on

all right so very good there’s a good example right there for revolving around

um horizon vertical axis so let’s do one now that is a little bit

different so let’s go back to and look at the next example here

so on this one right here we’re going to be um revolving um y equals natural log

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and the y axis and we’re going to be looking on this interval here

and we’re going to be revolving around the line x equals minus 1. so

this will give us another example where we’re doing the washer method and

revolving around a vertical axis right here

but not the y-axis all right so we’ll get one example of each type

all right so let’s uh sketch the graph here and see what see what’s going to

happen so the first thing i’m going to do is graph the natural log of x so it’s

coming right through there like that and now on the

y-axis we’re only going to be looking at these y values right here so 0 up to

about a one so i’m going to take this part here away we don’t need it and that

and so this is coming from the natural log of x and

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and it’s going to be okay so the region bounded by this and the y-axis so

what’s this height right here so this is coming through here right here

is going to be a and the height right here is going to be so this is like a 2

and then an e right here because natural log of e is 1 right and

so we’re going to get the height of 1 right there all right perfect

and so let’s just chop this off right through here

and the reason why is because we’re bounded less than one and we’re greater

than zero so we’re bounded right here also

so the y’s bounded between zero and one all right so let’s just shade this in

real quick we got this uh region right here that

we’re going to be revolving around the line x equals um one

00:37

and so yeah let’s sketch that in here let’s call this the y-axis the x-axis

and here’s about an x there’s about a one there so let’s call this about a

minus one here and let’s say that this is the vertical line here x

equals minus one and we’re going to be revolving around

this axis here this vertical axis right here x equals minus one

so we’re going to be getting this uh solid right here generated as we rotate

this around and so yeah we’re going to be using

washers when you rotate it around you’re

going to get a disc but it’s going to be

hollowed out so what you really get is a washer so when we revolve this around

here we’re getting washers all right and so we need to know

what is the uh r of y the outer radius so the r of y is 1 plus e to the y so

this right here is y equals natural log of x right

or if you solve for x it’s just x equals

00:38

e to the y right because natural log and e are inverse functions of each other

and so or if you want to think about it like

this e to the y is e to the natural log of x

but e to the natural log of x is just x well however you want to think about it

this is e to the x equals e to the y also so because we’re revolving around a

vertical axis right so this is my uh r of x here let’s put it in red

this is my r of x here so here to here and so this is going to be 1

and then plus the e to the y so this right here is e to the y right here

and so we pick a y value on the y axis and this distance is the

one and then plus the e to the y all right and so then for the lower

radius right here uh r of the y is just one and the reason why is because we’re

00:39

chopped off right here by the y-axis so this distance right here is just a one

all right and so now that we know the inner and outer radiuses we can go find

the volume so the volume will be pi i’m going to pull out my pi already 0 to 1

and then we’re going to have this first function right here the uh outer radius

1 plus e to the y squared minus and then the second one 1 and then

that squared and then d y so that’s a pi pi

all right so yeah we’re gonna integrate this out we’re gonna square this out

minus one we’re gonna expand that and simplify it and so this right here will be

um well i’ll just uh skip to the end here because we did this in a previous

episode so anyways the value that you get at the end is pi over 2 e squared

00:40

plus four e minus five all right so pi over two times e squared

plus four e minus five and so that’s the volume that you get of

that solid right there so i think setting this up by hand sketching the

region and understanding the outer radius and inner radius is

really nice to do by hand when you’re learning this

and then let’s go see how to do this in python now

so here we go let’s look at the python now for example 4 here

and so let me get rid of this real quick right here check this one out

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to get noticed and recognized anyways find the volume of the solid and by by

the way thank you for all the people who’ve already subscribed really

appreciate it i hope you enjoy the new episodes and look forward to them

all right so find the volume of solid generated

by natural log of x y axis and the interval is revolved about the line x

00:41

equals minus one so first thing i’m going to do is uh define the function

right here natural log of x now when you write in python it’s just logx

that’s already the natural log right there and i’m using the numpy the

you know the numpy package right there to do that

and then we’re going to make some axes of a figure customize and

you can play with the aspect ratio if you want but i commented out

all right and so then i’m going to make a bunch of x’s so i don’t want to pick

exactly 0 so i don’t want to start it at

zero that’ll give me an error so i chose something really really close to zero

and then i go to e so this will be like really close to zero all the way to e

which is about two point seven something so there’s my x’s i’m going to plug in

my x’s into y i’m going to plot that function right

00:42

there you can see it right here and then um now i’m going to plot the

this goes from one to e um and then the y’s again all right so i

did this right here so that i can get um this

let’s see here this is a bunch of x’s from 1 to e

all right so this is to help me shade so i want to fill this in here and shade

this region right here and then i want to shade this region right here so this

is 0 to one and then this is one two to e so i graph i graphed this one again

and i filled it in and i grabbed this one right here and i filled it in um

so that way i can shade it in so it’s a little difficult to not

i don’t want this to shade this end down here so that’s why i did it is to do

those two fill betweens i filling in here and i’m filling in here

and so that’s the region right there that’s filled in

00:43

all right so to integrate this i need to identify

my outer and inner radiuses so i’m going to use sci pi i’m going to integrate

using the quad method now i’m going to have an anonymous

function here it’s going to be a function of y so this will be um

pi times the e to the y plus 1 squared and then minus 1 squared

and then i’m going to integrate along the y-axis 0 to pi

and i’m going to take the numeric value from that all right so let’s click in

here and execute it there we go and so yeah this gives us the

approximation right here 20.832 and if you’re wondering how accurate

that is we can take that off there if you want to and then we’ll get not only

the numeric value but also how accurate it is

and gives you some indication of how accurate it is right there lots of zeros

in there so this is very accurate right here so you know

00:44

speaking for educational purposes that’s very accurate right there

and so that gives us the numeric approximation to the value that we had

if you remember the exact value was pi over two and then it was

e squared plus four e minus five all right and so that was the exact value

and then we found the approximate value right there

so yeah i want to say thank you for watching and look forward to seeing you

in the next episode and i’ll see you then if you enjoyed this video please like

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