Disc method (rotating f(x) about x axis) The volume of y=sqrt(x) between x=0 and x=1 rotated around x-axis
Disc method (rotating f(x) about x axis)
- Welcome back.
- On the last video we came to the conclusion that we could
- figure out the volume when we rotate a function about the
- x-axis, so let's apply that to an actual exercise.
- I'm going to erase everything because I don't want you to
- memorize this, because frankly I haven't memorized this.
- And if you do, you'll forget it one day and then you
- won't know how to do it.
- But if you understand why it works then you'll never forget.
- As long as you remember basic integration.
- Maybe you want to memorize it if your teacher tends to give
- you a test that don't have much extra time in them, just
- to speed up the process.
- But you should know what's going on.
- So let me draw the axes again.
- That's my y-axis.
- That's my x-axis.
- And so since our first example was y equals square root of
- x, let's stick with that.
- And for reasons that might become apparent that tends to
- be one of the more typical examples when you rotate
- things around axes.
- So let me see if I can draw it as well as I drew it last time.
- OK, so that's y equals square root of x, it's just f of x
- this time, I've defined it.
- This is the x-axis.
- That's the y-axis.
- And I'm going to rotate this around the x-axis again.
- So I'm going to get a sideways looking cup thing.
- And let's say I want to figure out the volume of that cup
- between the points 0-- and to make it simple, let's just
- say the points 0 and 1.
- So essentially we're just going to get a cup, a sideways
- cup, it's going to look something like this.
- It's going to look something like this.
- That's going to be the-- that's a horrible-- the
- opening of the cup.
- Actually why don't I use the circle tool.
- It just dawned on me that I had a circle tool.
- So the opening of the cup will look like that.
- Actually I could draw it right here.
- This would be the opening of the cup.
- Well-- you can see sometimes that my videos are
- a little unplanned.
- There you go.
- So that would be the opening of the cup.
- This is excellent.
- This tool is very well suited for what I'm doing here.
- We're rotating around that way.
- We're turning that function.
- So the cup's going to look like that, so the bottom part of the
- cup's going to look like this.
- And it's solid, so we want the volume of the whole thing.
- In future videos I'm going to show you actually how to figure
- out the surface area of the cup, which I find in some
- ways more interesting.
- So how do we think about that again?
- Let's just rederive it, but this time we'll use
- a specific equation.
- So we just have to figure out what is the volume of one disk
- and then sum up all the disks.
- So let's say this disk right here-- actually let's just take
- this disk at the end point right here that I've already
- drawn something for.
- So what's the radius of this disk?
- The radius of that disk is f of x at that point.
- Well f of x at that point is just square root of x.
- Radius is equal to square root of x.
- And so the area of that disk is going to equal pi r squared.
- Well the radius is square root of x, so it equals pi times
- square root of x squared.
- So it equals pi times x.
- That's the area of each disk.
- And then if we want the volume, you just have to multiply the
- area of that surface times the depth of the disk.
- I'm just trying to show.
- You can imagine that this is kind of like a quarter and this
- is the side of the quarter.
- We saw in the last video that depth, that's just a very
- small change in x, because we want each disk to be
- infinitesimally thin.
- So the width is just dx at any point.
- So the volume of each disk is equal to the area, which we
- just figured out was pi x times the depth, times dx.
- That's the volume of each disk.
- So the total volume is going to be equal to the
- sum of all of these.
- That was one disk I drew, then you're going to have another
- one here, you're going to have another one here,
- another one here.
- You're going to have infinitely many, and you want them to be
- super, super, super thin so that you get an accurate
- measure of the exact volume of this curve.
- Otherwise it would just be an approximation, and that's
- where we use the integral.
- So it will be the integral from.
- And my original boundaries were 0 to 1.
- The disk we used as an example, this is probably you know the
- last disk, so this one will actually have a radius of the
- square root of 1, which is 1.
- Not that you have to know that, I'm just trying to keep
- emphasizing the visualization.
- So what will be the integral?
- Well we're going to go from 0 to 1, and we're going to sum up
- a bunch of these disks, which we've already defined,
- so it's pi x dx.
- This is looking to be a fairly straightforward integral.
- So what's the integral of that?
- Pi is just a constant and the antiderivative of x is x to
- the 1/2 over-- I'm sorry.
- It's x squared over 2.
- I've been a little rusty since I last did some
- So we get x squared, we get pi times x squared over 2.
- That's the antiderivative of that.
- And then we have to evaluate it at 1 and then subtract
- it and evaluate it at 0.
- And so what do we have.
- We get 1/2 pi, so we get pi over 2 minus 0 pi, minus 0.
- So it equals pi over 2.
- There we go.
- We just figured out the volume of this cup from 0 to 1.
- Reasonably interesting.
- Let's see if we can do that again to figure out the-- just
- to give you another example, just hit the point home-- to
- see if we can figure out the volume of a sphere,
- the equation for the volume of a sphere.
- So what's the equation for a circle?
- It's x squared plus y squared is equal to r squared.
- And let's write that in terms of y is a function of x, just
- so we have something that we can work with the
- way we learned it.
- So we get y squared is equal to r squared minus x squared, and
- then we get y is equal to the square root of r squared
- minus x squared.
- Actually now that I realize it, I'm going to not do this,
- because I think I'm going into too complicated a problem.
- I did that on a fly.
- But in the next video I will do slightly more complicated
- without going to this one, because I probably don't have
- time for it I just realized.
- Anyway I'll see you in the next video.
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