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.