Volume of a bicylinder -- wait! What is a bicylinder?

[Music] here we're going to find the volume of something known as a double cylinder so a double cylinder is the intersection of two cylinders so i've put this set up into r3 so we've got the x-axis the y-axis and the z-axis and then running down the x-axis i have a cylinder of radius b and then running down the y-axis i have a cylinder of radius a and our goal is to find the volume of this intersecting object now my picture of this intersection isn't that great so i'll put a picture that's a little bit better on the screen right now just to give you guys an idea of what this looks like okay so now let's jump into trying to find this volume and we'll do this by slicing this with a plane which is like z equals z naught or z is fixed but in order to do that we'd probably like to write down equations for each of these cylinders so let's talk our way through how we can write down those pretty easily so let's look at this one first the one coming down the y axis so here we can think about the y component of every point on this cylinder being free to be anything that we want but then the x and the z components have to build a circle of radius a so that gives us some motivation for the equation of this cylinder as x squared plus z squared equals a squared so let's maybe color code that a bit so this is going to be the cylinder which is yellow and then similarly we can write down the equation of the cylinder radius b that's running down the x-axis so we'll think about it like this the x component is free to be anything but the y and the z components have to be along this circle so i can write it down as y squared plus z squared equals b squared again because it's a circle of radius b in this case so i'll draw my cylinder of the right color right there now like i said we're going to slice this with a horizontal plane a horizontal plane has the equation z equals a number so we'll slice it with maybe z equals z naught and see what the picture looks like so let's maybe write that down we're going to slice with the plane z equals z naught so that means that z is just fixed so it's fixed arbitrarily but it cannot change it's no longer variable okay so that means this equation right here will collapse to x squared plus z naught squared equals a squared but i'll write that as x squared equals a squared minus z naught squared keeping in mind that this number over here is now a constant let's remind ourselves of that this is a constant at this point because a is always a constant and z naught has been fixed then similarly over here we'll get y squared equals b squared minus z naught squared now we can take the square root of both sides and we'll see that we get x equals plus minus the square root of a squared minus z naught squared and we'll get y equals plus minus the square root of b squared minus z naught squared now we can think about drawing a picture of this in the plane because notice what we have here is really just x is fixed to be positive or negative the same number and y is also fixed to be positive or negative the same number so here i'll draw this x y plane but just keep in mind that this isn't really the x y plane this is the copy of the x y plane where z has been fixed at z naught so it's lifted up a bit or lifted down a bit so let's color code this result so here we have x is either plus or minus this number right here so plus that number would be like right here and minus that number would be like right there so here i'll put the square root of a squared minus z naught squared then i won't put minus that but we can just remember that that's the case so x will always have that value so if x is always this value that creates a vertical line now we can similarly do the same thing for y so maybe we'll put here this is y equals the square root of b squared minus z naught squared so if y is fixed there then that represents a horizontal line so i'll draw it like this and then the negative copy of that would be like down here so i'll draw that here but look what we've got we've got a rectangle and the side length is 2 times the square root of a squared minus z naught squared by 2 times that 2 because you know we've got we're going from negative that number to positive that number for the x and the y axis let's maybe be clear of that so this is a rectangle that is 2 square root of a squared minus z naught squared by 2 times the square root of b squared minus z naught squared great but it's easy to calculate the area of a rectangle so the area of this rectangle maybe i'll put a sub z naught because it depends on this z naught point will be equal to 4 times the square root of a squared minus z naught squared and then b squared minus z naught squared so it's getting kind of messy right there but we'll clean it up when we get it to the next board okay then how do we calculate the volume if this is the area of one of our slices well we'll just integrate that area from the smallest z value to the largest z value so using symmetry we can take 0 to be the smallest possible z value and then just double the volume that we get because we're looking at the volume just above the z axis we'll have an exactly the same volume below the z-axis so we'll take z starting at zero and then it will end at whichever one is smaller a or b let's maybe just say that we have an ordering here where a is less than b although doing it the other way would be fine as well so that means we'll have z run from 0 to a notice if it ran bigger than a then we'd end up with minus signs under the square root here and that wouldn't make much sense so let's maybe summarize what we have at the top of the next board and then we can write down an integral which will calculate the following volume so on the last board we figured out if we sliced our picture at a fixed z value we ended up with a rectangle and we calculated that area of that rectangle to be the following function that's in terms of z so it's 4 square root of a squared minus z squared times b squared minus z squared so that means we can easily calculate our volume so our volume will be the integral of this really as z goes from negative a to positive a but again like we hinted at on the last board this is an even function so that means that we can just double it and then integrate from 0 to a so if we double it and then also multiply by 4 that will give us 8 times the integral from 0 up to a of the square root of a squared minus z squared times b squared minus z squared okay nice now we have that integral to calculate but unfortunately i've got some bad news about this integral there's no closed form for the antiderivative so we can't express this any more simply than it is right now this is a so-called elliptical integral and i'm actually tossing around the idea of making a mini series about elliptical integrals as like afternoon releases let me know if you guys want to see something like that but if a is equal to b we can simplify it so if a is not equal to b then we can't simplify it we've got one of those elliptical integrals but if a is equal to b well it simplifies quite nicely so let's look at the special case when a is equal to b so i'll write that here um then we'll have the volume is equal to eight and then the integral from zero to a of a squared minus z squared d z because those two are the same so the square and the square root cancel each other but now this is just a polynomial function it's fairly easy to take the antiderivative so we'll have eight and then a squared times z minus z cubed over 3 we need to evaluate that up from 0 to a so evaluating it at the upper end point will give us 16 a cubed over 3 then evaluating the lower end point will give us 0. so that's the volume in the case when a is equal to b and like i said if a is not equal to b this is something called an elliptic integral and interestingly elliptic integrals have a lot of applications in modern romanjan style number theory so like i said i'm thinking about doing a mini-series on these types of integrals let me know if you'd like to see that and that's a good place to stop

Channel: Michael Penn

Views: 23,292

Keywords: math, mathematics, number theory, abstract algebra, calculus, differential equations, Randolph College, randolph, Michael Penn

Id: fz5H8BiPJGo

Length: 10min 18sec (618 seconds)

Published: Sat Dec 18 2021