Module 9 : Integral Calculus

903 : Solids of Revolution

O B J E C T I V E

In this module, we will examine solids created by rotating functions around the x and y axes, creating three dimensional colored graphs and computing these volumes symbolically, with numerical results.

S E T U P

In this project we will use the following command packages. Type and execute this line before begining the project below. If you re-enter the worksheet for this project, be sure to re-execute this statement before jumping to any point in the worksheet.

> restart; with(plots):

Warning, the name changecoords has been redefined

Note that this project creates many large graphs. Depending on the capabilities of computer system you may experience memory problems. You can overcome these in two ways. Whenever you plan to save your file, first remove the output. [tell them how to do this!!] When you re-enter your program, do not execute worksheet, but rather just re-execute the commands restart; with(plots): and then go to where you want to continue working.

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A. Revolving Around The X Axis

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Lets see what it looks like to revolve functions around the x-axis. First lets define a function and see what its graph looks like.

> f := x -> 2 - sin(x);

> plot( f(x),x = 0..2*Pi, y = 0..3);

We will revolve this curve around the x-axis. To create a three dimensional view of the resulting solid of revolution, we will use three dimensional coordinates (x,y,z). The z axis will measure how high off of the xy plane the surface is. Any given point on the original graph (x,f(x)) will be revolved around the x-axis to create a circle of radius f(x). Each point on the circle is of the form (a,b,c) where a = x, b2 + c2 = f(x)2. See the diagram below.

This can be a little confusing so dont worry about it too much.

Using this idea, we can revolve a function around the x-axis, by using this Maple command :

> implicitplot3d( {y^2 + z^2 = f(x)^2}, x = 0..2*Pi, y = -3..3, z =-3..3,
style = patchnogrid, orientation = [-50,70], axes = boxed );

Click anywhere on the graph and you will see a black border appear. While holding the mouse down and dragging you will see the graph rotate in three dimensional space. Please inspect this graph from all angles. You can look over, under, and through it!

Here is another example which you can use as a template for the exercise that follows.

> f:= x -> 1-x; a:= -1; b := 1;

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

> implicitplot3d( y^2 + z^2 = f(x)^2, x = a..b, y = -2..2, z =-2..2,
style = patchnogrid, orientation = [-50,70], axes = boxed, numpoints = 2000);

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B. Revolving Around The Y Axis

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We can also revolve curves around the y axis. However to accomplish this in Maple, we need to alter our scheme slightly to create a function x = g(y) which happens to be the inverse function of y = f(x). For example to rotate y = x2 around the y-axis, we write x = g(y) = sqrt(y).

> g := y -> sqrt(y);

> implicitplot( x = g(y),x= -1..1,y = -1..1);

> implicitplot3d(x^2 + z^2 = g(y)^2,x =-1..1, y =0..1, z=-1..1,
style = patchnogrid, orientation = [-90,-20], axes = normal, numpoints = 2000 );

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C. Revolving A Region Between Curves

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We can also revolve a region which is between two different functions. This is slightly more complicated. Here is an example.

> f := x -> x^3; g := x -> x^(1/3); plot( { f(x), g(x) }, x = 0..1);

> display(implicitplot3d(y^2 + z^2 = f(x)^2, x = 0..1, y = -1..1, z = -1..1,
style =patchcontour, shading = xyz),
implicitplot3d( y^2 + z^2 = g(x)^2, x = 0..1, y = -1..1, z = -1..1,
style =patchnogrid , shading = z));

In this display you can see that the inner and outer functions are shaded differently and by turning the graph you can get a visual idea of the thickness of the walls of the cup you see.

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D. The Disk Method

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In this section well compute the volume generated by revolving a function around the x-axis using the disk method. In this method, we imagine that we cut the area to be rotated into thin vertical slices of size y by delta x, rotate these slices to form a disk which you might think of a thin wafer or coin, then sum these up using integration. The formula is V = y2 dx.

Here is an example. We will rotate y = 1 + x*sin(x) between x = /4 and around the x axis.

> f:= x -> 1 + x*sin(x); a := Pi/4; b := Pi;

> plot( {f(x),0}, x = a..b);

Set up the integral, find its symbolic value, and its decimal value.

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

> value(%);

> evalf(%);

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E. The Washer Method

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The disk method we saw in the last section works only if the slice being rotated continues without interruption from the x-axis to the curve, and wont work if there are gaps. Consider this example where we plan to rotate the area between these two functions around the x-axis

Vertical slices intersect the area to be rotated between these functions but do not continue down to the x-axis.

> f := x -> x^(1/8);

> g := x -> x^8;

> a := 0; b := 1;

> plot( {f(x), g(x)}, x = a..b);

We thus need to use washers rather than disks. The formula is V = (f(x)2 - g(x)2) d. It is straightforward to implement this in Maple. Set up the integral, find its symbolic value, and its decimal value.

> Int( Pi*(f(x)^2-g(x)^2), x = a..b);

> value(%);

> evalf(%);

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F. The Cylindrical Shell Method

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In this section well compute the volume created by rotating a function around the y axis using the cylindrical shell method. The formula is V = 2xy dx. Here is an example.

> f:= x -> sqrt(abs(x^2 - 2 )); a := 0;b:= 4;

> plot( {f(x),0}, x = a..b);

Note that it would be very difficult to use the washer method with this problem even using Maple. You would need to take horizontal slices and the volume would be something like (42 - s2 + r2) where r and s would both be determined by the function. Fortunately, it is much easier using the shell method.

Set up the integral, find its symbolic value, and its decimal value.

> Int( 2*Pi*x*f(x), x = a..b);

> value(%);

> evalf(%);

>