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The purpose of this lab is to use Maple to study planar domains and the solids of revolution. Solids of revolution are created by rotating curves in the x-y plane about an axis, generating a three dimensional object. They are discussed in Chapter 6 of Calculus by Varberg and Pursell (sections 2 and 3). The specific properties of them that we wish to study are their volume, surface area, and graph.
If we have a curve lying entirely on or above the x-axis, which is the
graph of a function y = f(x), and x is restricted to being between a
and b, then we may rotate it about the x-axis, generating a three
dimensional object, a solid of revolution.
For example, if the curve is simply f(x) = ax/h (a straight line) and x is between and h, we obtain a cone of height h and radius a.
Consider f(x) = x2 + 1 for which appears in Figure 1.
If we take the region between the graph of f(x) and the x-axis and
revolve it about the x-axis then it becomes the surface of the solid
pictured in Figure 2.
If f(x) = x1/2 and x is between, say, and 1, we obtain a bowl
shaped object called a paraboloid. (Note the equation y = x1/2 is equivalent to x = y2, a parabola).
If f(x) = (a2 - x2)1/2 and x is between -a and a, we obtain a sphere of radius a (since the equation y = (a2 - x2)1/2 is equivalent to x2 + y2 = a2, a circle of radius a).
The area of the region R below the graph of the function , and above the x-axis is given by
(see p. 263). The region R is bounded by the curve involving the curvilinear part the horizontal base [a,b] with length b - a, and two vertical segments with lengths f(a) and f(b). The length P of the curvilinear part of the boundary is given by
(see p. 321).
The volume of the solid obtained by rotating the curve about the x-axis is given by
(see Example 4a on page 316 of the textbook for illustration)
The volume is obtained by adding up volumes of cylindrical disks, while its surface area (p. 324) is given by
Note that both of these formulas are for rotating the graph of f(x) about the x-axis.
We will consider planar domains defined as a union of the
region R introduced above and the region produced by
reflection of R in the x-axis. Clearly, the area of equals 2A, and the perimeter of equals 2[P + f(a) +
f(b)]. Particularly, if the region R is defined as the interior
of a half of the ellipse
belonging to the upper half plane y > 0 then becomes the interior of a half of the reflected ellipse
belonging to the lower half plane y < 0. So, - a complete ellipse, and its area is equal to 2A, whereas its perimeter equals 2P since f(a) = f(b) = 0 in this case.
The area radius of the planar domain is defined as
the radius of the circle possessing the same area as , i.e.
The perimeter radius of the planar domain is defined as the radius of the circle having the same perimeter as , i.e.
The volume radius of the body is defined as the
radius of the sphere possessing the volume V, i.e.
The surface radius of the body is defined as the radius of the sphere having the surface area S, i.e.
To plot the graph of the function y = f(x) given by the list of points we use the command plot in the relevant format, e.g.
> plot([[5,5],[6,7],[7,10],[8,12]]);Check it!
To do graphics in three dimensions, we utilize the tubeplot
function from the plots library of Maple, along with suitable
For sake of demonstration, suppose we wish to plot the surface of revolution one gets from rotating the curve about the x-axis. Maple commands to accomplish this would be:
> f:= x -> sqrt(x) + 1;
> with(plots): PltStyle:= axes= NORMAL, style= PATCHNOGRID, tubepoints=40;
> tubeplot([x,0,0], x=0..9, radius=f(x),PltStyle);
The last command uses the function tubeplot from the plots library, rotating about the x-axis (as controlled by the first argument, [x,0,0], letting x range from to 9, using the function f(x) to determine the radius at each point, and setting the options as set up by the line assigned PltStyle). The only especially noteworthy thing set by PltStyle is the last option, tubepoints. This effectively determines the resolution of the graph; the higher the value, the smoother it is (but the longer it takes). A value as low as 10 gives a fairly crude surface, by comparison. (Try it!)
Since this involves straightforward integration, one may simply issue commands such as:
> f:= x-> sqrt(x) +1;
> vol:= Int(Pi*f(x)^2, x=0..9);
Assuming the function has been defined, one might issue:
> S:= Int(2*Pi*f(x)*sqrt(1+D(f)(x)^2), x= 0..9);
For the function,
the graph in the (x,y) plane is a half of ellipse; revolving this graph about the x-axis, we generate an ellipsoid of revolution.
Jane E Bouchard