Welcome to a video on the surfaces of revolution.
The goals of this video are to determine the equation of a graph that is revolved about the X, Y, or Z axis, and then also to graph the equation of a surface of revolution.
Here's the idea, a surface can be generated by revolving a radius function r in two variables about one of the axes.
The equation of the surface is given by one of the following equations based upon which axis the function is rotated about.
We have y squared plus z squared equals r of x squared if it's rotated about the x-axis.
We have x squared plus z squared equals r of y squared if it's rotated about the y-axis.
And we have x squared plus y squared equals r of z squared if it's rotated about the z-axis.
If you look at the left side of these equations, they should remind you of the equation of a circle.
So really what's happening here, is we have these circles that are projected along the x, y, or z axis where the function controls the radius of those circles.
Let's take a look at an animation to get a better idea of what's happening here.
If we take a curve in two dimensions and rotate it about one of the axes, it would produce a surface as we see here.
Now if you take a look at just one strip of this surface, you can see it looks like a circle,
and the radius of this circle would be controlled by the function value of the curve that's rotated about the given axis.
And this is where those equations are coming from.
Let's go and take a look at an example.
Here we want to determine the equation of z equals the square root of x rotated about the x-axis.
Since we know it's rotated about the x-axis, we're going to be dealing with the equation y squared plus z squared equals r of x squared.
Let's first graph this function in the x-z plane.
So when x is zero, z would be zero.
When x is one, z would be one.
When x is four, z would be two.
So we're taking this function in the x-z plane and rotating it about the x-axis.
So we would rotate it about this axis here, producing a surface.
So the equation of that surface would be y squared plus z squared equals r of x squared.
Well r of x is just the square root of x.
So we would have y squared plus z squared equals the square root of x squared.
So the equation of this three-dimensional surface would be y squared plus z squared equals x.
Let's see if we can make a sketch of this before we take a look at it using some graphing software.
Again, we're going to graph this in the x-z plane.
So we have a point at the origin.
And then we have a point at one, one.
Somewhere here.  And then we have a point four, two.
Somewhere in here.
So on the x-z plane, the graph would look like this.
If we rotate it about the x-axis, it would create a surface.
That would look something like this.
Here's a graph of it using some graphing software.
And again, here's the generating curve.
Let's go and take a look at another one.
Notice here, we're going to be rotating the same function, but now we're gonna rotate it about the z-axis.
So because it's about the z-axis, we know that the radius function must be a function of z
and the other two variables would be on the left side, x squared plus y squared.
Let's first go ahead and sketch this graph again.
But now we'll be rotating about the z-axis so it's going to produce a different surface.
Now the equation of this surface will obviously be different.
Notice that the radius function has to be r of z which means we have to solve this equation for x so that we can have a function in terms of z.
So if we take z equals the square root of x
and we square both sides of the equation,
we're gonna have z squared equals x which means r of z is going to be z squared.
So the equation of this surface will be x squared plus y squared equals z squared, squared
or x squared plus y squared equals z to the fourth.
So you do have to be careful to make sure that you have the equation given solved for the correct variable, this has to be a function in terms of z.
Meaning we have to solve this for x so that we have a function in terms of z.
So if we sketch this generating curve on the x-z plane, it would look something like this, just like the previous video.
And if we rotate this about the z-axis this time, it would produce a surface that looks something like this.
Again, here we see the generating curve rotated about the z-axis and it produces this shape here.
So not only do we have the graph of this surface, we also were able to find the equation of this surface pretty easily.
I hope you found this video helpful, thank you for watching.
