Solids of Rotation

Prerequisites: Riemann Sums, The definite integral as an area under a curve.

Goal: To find a way to evaluate the volume of solids of rotation.

Today we begin by starting Maple, then copying and pasting the following code into our Maple worksheet. The color is only for your own reference; you may want to edit the purple characters, but probably not the red.

restart: with(plots):
a := -1:         # left endpoint
b :=  2:         # right endpoint
f := x -> x^2+1: # the function
plot(f,a..b,scaling=constrained, view=[-1..2,0..5]);
p1 := plot3d([x,f(x)*cos(t),f(x)*sin(t)],x=-1..2,t=0..2*Pi,grid=[16,24]):
p2 := polygonplot3d([seq([a,f(a)*cos(2*Pi*t/24),f(a)*sin(2*Pi*t/24)],
        t=0..23)]):
p3 := polygonplot3d([seq([b,f(b)*cos(2*Pi*t/24),f(b)*sin(2*Pi*t/24)],
        t=0..23)]):
display(p1,p2,p3,scaling=constrained);

After pasting this code into Maple, press <enter> (or <return>). The graph of the function f(x)=x²+1 should appear from x=-1 to x=2. Imagine taking this region and rotating it around the x-axis to obtain a solid. That solid should also appear below the orignal graph; reorient it (using the point-and-click interface) so that its relationship to the graph above it is clear. Our first goal is to find the volume of the solid.

restart: with(plots): with(plottools):
a := -1:         # left endpoint
b :=  2:         # right endpoint
f := x -> x^2+1: # the function
n :=  5:         # number of subintervals
dx:=(b-a)/n:     # look familiar?
p := seq(rotate(cylinder([0,0,a+i*dx],f(a+i*dx),dx),
         0,3*Pi/2,0),i=0..n-1):
display(p,scaling=constrained,axes=normal);
Again cut and paste this code into Maple and press <enter> (or <return>). This time, instead of the solid, you should get a picture of 5 cylinders that approximate it.
  1. Using the point-and-click interface, reorient the solid shape and the collection of cylinders so that their relationship to the first graph is clear.
  2. What is the width (thickness) of each cylinder?
  3. What is the radius of the first cylinder? The second? The third? The fourth? The last?
  4. What is the volume of the first cylinder? The second? The third? The fourth? The last?
  5. The volume of the solid is thus approximately the sum of these five volumes. What is it?
  6. To find a more accurate approximation, change the number n in the above code to a 9, and repeat questions 2-5.
  7. To find a more accurate approximation, change n to 27. Since it could become tiresome to do each radius and volume separately, it will help to find a general formula for the radius and volume of the ith cylinder. What is the radius of the ith cylinder? What is the volume of the ith cylinder? What is the total volume of the cylinders?
  8. Find a general formula for the radius and volume of the ith cylinder for arbitrary n. What is the total volume of the cylinders?
  9. To find the exact volume, we need to take a limit. How does that work? Find the exact volume of the solid.
  10. Repeat the problem for the function f(x)=2x-1 from x=1 to x=3.
  11. Check your work in problem 10 by noticing that the solid is just a cone with the tip chopped off, and finding the volume using geometry.
Last Update: 2007/06/21