Surfaces of Revolution

This demonstration was put together using MenuMaple.  The current version in the lab in Math 206/208 supports these commands which are found by using the plot... button on the calculus menu.

Here is the expression we will use.

> f:=sin(x);

Look at the graph generated by revolving f with x between 0 and Pi about the line y = 0.

> rotate(f,x=0..Pi,y=0,1);

The following shows several steps in the rotation of f with x between 0 and Pi about the line y = 0.  The first picture in the sequence shows f graphed over the interval from 0 to Pi.  The rest of the slides in the animation complete the revolution of f about y = 0.

> rotate(f,x=0..Pi,y=0,2);

The volume can be computed using the disk method. The next movie shows 8 disks being used. What would the picture look like if we could use an infinite number of disks?

> diskdemo(f,x=0..Pi,y=0,8,style=wireframe);

The radius of each disk used is the value of f at the left endpoint of a partition of the interval from 0 to Pi. The area of each disk is then Pi * f ^2. Using the left sum command on the Riemann...we get the following estimate of the volume.

> L8:=Lsum(Pi*f^2,x=0..Pi,8);

This simplifies.

> L8:=simplify(L8);

The definite integral which computes the volume is

> Int(Pi*f^2,x=0..Pi);

Evaluating this we have

> actualAns:=int(Pi*f^2,x=0..Pi);

With the rotate command we can revolve the same region used above about the line x = -2.

> rotate(f,x=0..Pi,x=-2,2);