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So far we have used the integral mainly to to compute areas of plane regions. It turns out that the definite integral can also be used to calculate the volumes of certain types of three-dimensional solids. The class of solids we will consider in this lab are called Solids of Revolution because they can be obtained by revolving a plane region about an axis.
As a simple example, consider the graph of the function f(x) = x2+1 for , which appears in Figure 1.
To help you in plotting surfaces of revolution, A Maple procedure called revolve has been written. The command used to produce the graphs in Figures 1 and 2 is shown below. The revolve procedure, as well as the RevInt, LeftInt, and LeftDisk procedures described below are all part of the CalcP package, which must be loaded first. The last line in the example below shows the optional argument for revolving the graph of f(x) about the line y=-2 instead of the default y=0.
> f := x -> x^2+1;
The revolve command has other options that you should read about in the help screen. For example, you can speed the command up by only plotting the surface generated by revolving the curve with the nocap argument, and you can also plot a solid of revolution formed by revolving the area between two functions. Try the following examples. (Note: The last example shows how to use revolve with a function defined piecewise.)
It turns out that the volume of the solid obtained by revolving the region in Figure 1 between the graph and the x-axis about the x-axis can be determined from the integral
to have the value . More generally, if you revolve the area under the graph of g(x) for about the x-axis, the volume is given by
Where does this formula come from? To help you understand it, Two more Maple procedures, RevInt and LeftDisk, have been written. The procedure RevInt sets up the integral for the volume of a solid of revolution, as shown below. The Maple commands evalf and value can be used to obtain a numerical or analytical value.
> f:= x-> sqrt(x) +1;
> vol:= int(Pi*f(x)^2, x=0..9);
> S:= int(2*Pi*f(x)*sqrt(1+D(f)(x)^2), x= 0..9);
Once you have generated the volume for this equation, please plot the graph of the solid, using a radius =2 ft and a height = 5 ft. What geometric object is this?
Once you have generated the volume for this equation, please plot the graph of the solid, centered around the origin, with r=12 ft.
Find the volume of the solid generated by this equation. Plot the solid formed when a=3 and b=7. What geometric object is this?
x2 + (y-a)2=r2about the x-axis. Find the volume of the torus generated by this equation. After finding the general form of the volume of a torus, plot this solid by substituting numerical values for a and r. Note: In order to create a torus, r<a. Why must this be true?
Christine M Palmer