> f := x-> x^2+1; > plot(f(x),x=-2..2);If we take the region between the graph and the x-axis and revolve it about the x-axis, we obtain a 3-D solid.

To plot this solid of revolution using Maple, the `revolve` command can be used. All of the procedures described in this lab are part of the `CalcP7` package, which must be loaded first. The syntax for `revolve` when revolving about the -axis over the interval
is:

> revolve(f(x),x=a..b);If you revolve the area under the graph of for about the -axis, the volume is given by

The integral formula given above for the volume of a solid of revolution comes, as usual, from a limit process. Recall the rectangular approximations we used for plane regions. If you think of taking one of the rectangles and revolving it about the x-axis, you get a disk whose radius is the height of the rectangle and thickness is , the width of the rectangle. The volume of this disk is . If you revolve all of the rectangles in the rectangular approximation about the x-axis, you get a solid made up of disks that approximates the volume of the solid of revolution obtained by revolving the plane region about the x-axis.

To help you visualize this approximation of the volume by disks, the
`LeftDisk` procedure has been written. The syntax for this command is similar to that for `revolve`, except that the number of
subintervals must be specified. The examples below produce
approximations with five and ten disks. The last example produces the exact solid of revolution.

> with(CalcP7): > f := x -> x^2+1; > LeftDisk(f(x),x=-2..2,5); > LeftDisk(f(x),x=-2..2,10); > revolve(f(x),x=-2..2);The

> revolve(f(x),x=-2..2,y=-2,nocap);You can also plot a solid of revolution formed by revolving the area between two functions. Try the following examples.

> plot({5,x^2+1},x=-2..2); > revolve({5,x^2+1},x=-2..2);

Therefore, the volume of the solid obtained by revolving the region between the graph of and the -axis about the -axis can be determined from the integral .

In order to calculate the volume of a solid of revolution, you can either use the `int` command implementing the formula above or use the Maple procedure `RevInt` which sets up the integral for you. The Maple commands `evalf`
and `value` can be used to obtain a numerical or analytical value. Approximations to the volume of the solid of revolution can be found using the `LeftInt`. Try the examples below to see the different types of output.

> Pi*int(f(x)^2,x=-2..2); > evalf(Pi*int(f(x)^2,x=-2..2)); > RevInt(f(x),x=-2..2); > value(RevInt(f(x),x=-2..2)); > evalf(RevInt(f(x),x=-2..2)); > evalf(LeftInt(f(x),x=-2..2,50));

- For the function
over the interval
,
- Plot over the given interval.
- Plot the approximation of the solid of revolution using with 12 disks.
- Plot the solid formed by revolving about the -axis.
- Plot the solid formed by revolving about the line .
- Find the exact volume of the solid of revolution about the -axis using the integral formula.
- Find the exact volume of the solid of revolution using the
`RevInt`command and label your output`exact`. - Find the minimum number of subintervals needed to approximate the volume of the solid of revolution about the -axis using
`LeftInt`with error no greater than 0.01.

- What function's graph can be revolved about the -axis to obtain a sphere of radius ? Use this function and the
`RevInt`command to prove that the volume of a sphere is . - Six years ago, Kevin Nordberg and James Rush (both
class of '98) were asked to design a drinking glass by revolving a
suitable function about the axis. Here is the function they came
up with.

They obtained the shape of their glass by revolving this function about the axis over the interval . The Maple command they used to define this function is given below.> f := x -> piecewise(x<-3/4,-2*x-1/2,x<0,1/6,x^(2/3)+1/6);

Plot this function (without revolving it) over the interval and identify the formula for each part of the graph. Then, revolve this function about the axis over the same interval and comment on the glass Kevin and Jim designed. Finally, compute the volume of the part of this glass that could be filled with liquid, assuming the stem is solid. (Hint - your integral will involve only one of the formulas used to define the function.)

2001-11-27