Revolution about the y-axis:

Consider the lamina bounded by the graphs of y=f(x) and y=g(x) from x=a to x=b.  We'll revolve this lamina about the y-axis.

>

restart:

>	f:=x->(x+7)^2-5*(x+7);

>	g:=x->2*(x+7);

We first plot the graph, using "scaling=constrained" to force the units on x- and y-axes to be equal.:

>	a:=-7;b:=0;

We plot the graphs.

>	plot([f(x),g(x)],x=a..b,scaling=constrained);

>

If we revolve these graphs about the y-axis, we obtain a surface of revolution.  This surface is the graph of a function of two variables defined on a portion of a horizontal plane.  We obtain the graph of the resulting surface by using what are called cylindrical coordinates.  To get cylindrical coordinates, we revolve the x-axis in a circle.  When the x-axis has been revolved an angle   , a point that initially was on the x-axis at x-coordinate x=c has been rotated to lie on the rotated x-axis at a distance from the origin equal to the absolute value of c.   We say that, in cylindrical coordinates, the point is at position  and an angle of   .   The height of a point on the surface above this point is f(r).   We add a "z_cylindrical coordinate system"  to Maple's library of coordinate systems with the following command:

>	readlib(addcoords)(z_cylindrical,[z,r,theta],[r*cos(theta),r*sin(theta),z]);

Now plot the surface obtained by revolving the graph of f(x) about the y-axis:

>	plot3d(f(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical);

>

If you  click anywhere in the picture, you can grab the surface with your mouse and twist it to obtain any view you desire.  If you click on the picture with your right mouse button, you can select various styles. Wireframe  style allows you to see hidden portions of the surface.  Experiment with various choices.  You can change the color by adding the option color=red , for example, to the list of options after coords=z_cylindrical. .  If your surface is one color, you will probably want to highlight it by adding lightmodel=light4 , for example to the list of options.   Possible lightmodels are light1 through light4.

Now we'll look at the solid generated by revolving the lamina bounded by the graph of   , , , and the graph of g(x)=2*(x+7) about the -axis.

First sketch the lamina:

>	a1:=plot(f(x),x=a..b,color=navy,thickness=2):

>	a2:=plot(g(x),x=a..b,color=navy,thickness=2):

We load the plots and the plottools programs:

>	with(plots):with(plottools):

and plot their graphs:

>	display([a1,a2],scaling=constrained);

Now we will revolve this region by plotting the surface of revolution generated by each of the two boundary curves:

>	p1:=plot3d(f(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical):

>	p2:=plot3d(g(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical):

>	with(plots):display([p1,p2]);

>

Revolution about the x-axis:

To show the solid generated by revolving the region bounded by y=x*sin(x)+12 and y=x*(x-2*Pi)+12 about the x-axis:  Here we use a command called tubeplot , which sketches a tube about the x-axis whose radius at any point is the height of the function.

>

restart:

>	f:=x->x*sin(x)+12;

>	g:=x->x*(x-2*Pi)+12;

>	a:=0;b:=2*Pi;

We load the plots and the plottools programs:

>	with(plots):with(plottools):

>	plot([f(x),g(x)],x=a..b);

>	t1:=tubeplot([0,x,0,x=a..b,radius=f(x)]):

>	t2:=tubeplot([0,x,0,x=a..b,radius=g(x)]):

>	display([t1,t2]);

>

Here again, you can rotate the figure to any desired position, change the style, color, axes, or perspective.

Your own examples:

Now you can create your own surfaces of revolution.  Simply define one or more functions f, g, h, etc., determine the domain [a,b] that you want to display, and enter the commands.  You need to understand that a and b represent values of r; they are the least and the greatest distance  from x=0 in your domain, so they must be non-negative numbers.  For example, if your curve f(x) is to run from x=-7 to x=-3, let r run from a=+3 to b=+7.   Experiment to see if you can obtain some interesting surfaces.  You might try, for example, to show a torus, a sphere with a hole drilled through the center, or some other interesting solid.

>

restart:

>

f:=x->  ;

>

g:=x-> ;

You can define a third function h, fourth function k, and so on.

>

a:=  ;

>

b:=  ;

Sketch the curve or lamina obtained by revolving about the y-axis:

>	plot([f(x),g(x)],x=a..b,scaling=constrained);

>	### WARNING: persistent store makes one-argument readlib obsolete readlib(addcoords)(z_cylindrical,[z,r,theta],[r*cos(theta),r*sin(theta),z]);

>	with(plots):with(plottools):

>	p1:=plot3d(f(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical):

>	p2:=plot3d(g(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical):

If you defined a third function, define a p3 (by copying and pasting the p1 command and changing p1 to p3, f to h) and add it to the list in the following command.

>	display([p1,p2]);

>

Now revolve about the x-axis:

>	with(plots):with(plottools):

>	t1:=tubeplot([0,x,0,x=a..b,radius=f(x)]):

>	t2:=tubeplot([0,x,0,x=a..b,radius=g(x)]):

>	display([t1,t2]);

>

Revolution about a line other than the x- or y-axis:

If you want to revolve your curve or lamina about a vertical line other than the y-axis and display the surface or solid that is generated, translate your functions the appropriate distance to the right or left and revolve about the y-axis.  For example, in order to revolve about the line x=-5, translate your functions 5 units to the right by replacing each occurrence of x with x-5.  

If you want to revolve your lamina about a horizontal line other than the x-axis, translate up or down.  For example, to revolve about the line y=5, translate your functions down by replacing y with y+5.  Then revolve about the -axis.

Project:  Solids of Revolution

• A:  Let f(x) = sin(x) on the interval  and let R  denote the lamina between the graph of f  and the x -axis.
• 1.  Display the solid of revolution obtained by revolving the lamina R  about the x -axis. Determine the volume of the solid of revolution.
• 2 . Display the solid of revolution obtained by revolving the lamina R  about the y- axis. Determine the volume of the solid of revolution.
• 3.  Display the solid of revolution obtained by revolving the lamina R  about the vertical line x  = -1. Determine the volume of the solid of revolution.
• B:   Let m and n be the number of letters in your first and last names, respectively.  Create a torus (a doughnut) of inner radius n  by revolving the graph of a circle of radius m centered at (m+n,0) about the y-axis.  
• 1.   Define and display the circle you are revolving, then display the torus.
• 2.  Fnd the volume of the torus by describing the method you are using (slice or shell),  setting up the appropriate integral (be sure to explain to the reader each element of your integrand), then having Maple compute the value of the integral.
• C:   Create two interesting solids of revolution by revolving a lamina bounded by the graphs of two functions about the y-axis then about the x-axis.  Use at least one trigonometric, logarithmic, or exponential function like those given in the note below for part of the boundary of the lamina.  Find the volume of each solid.  Be sure to:
• 1.  Describe the lamina that you are revolving and sketch its graph.  
• 2.  Display the solid.  Be sure to remove any flanges.
• 3.  Calculate the volume of the solid.  Be sure to describe the method you are using (slice or cylindrical shells), set up the appropriate integral, explain each element of your integrand, then have Maple compute the value of the integral.

• Note:  Some interesting functions you can use include f(x)=x+sin(x),f(x)=x^2*exp(-x), f(x)=ln(x)/x, f(x)=arctan(x), in combination with polynomial functions. Choose your intervals carefully so that you do not have flanges and your functions are defined throughout the interval.

Tools for this Project:

• Note:   A good way to begin working on your project is by copying and pasting the Tools for this Project into a new file, then inserting your expository paragraphs before, between, and after the Maple input and output statements.  Be sure to begin the solution of your problem with a statement of that problem, to explain each step of your work (in particular, the reader should be able to understand your formulation of each integral without having to work the problem for himself or herself), and to end your report with a concluding paragraph.

Parts A and B:

Sketch the surface obtained by revolving a lamina bounded by the graphs of f and g about the y-axis:

>	### WARNING: persistent store makes one-argument readlib obsolete readlib(addcoords)(z_cylindrical,[z,r,theta],[r*cos(theta),r*sin(theta),z]);

>	with(plots):with(plottools):

>	p1:=plot3d(f(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical):

>	p2:=plot3d(g(r),r=a..b,theta=0..2*Pi,coords=z_cylindrical):

>	display([p1,p2]);

Part B:

Sketch the surface obtained by revolving a lamina bounded by the graphs of f and g about the x-axis:

>	with(plots):with(plottools):

>	plot([f(x),g(x)],x=a..b);

>	t1:=tubeplot([0,x,0,x=a..b,radius=f(x)]):

>	t2:=tubeplot([0,x,0,x=a..b,radius=g(x)]):

>	display([t1,t2]);