This is an interesting exercise in setting up triple integrals in cylindrical and spherical coordinates to obtain a formula that is immediate without any calculation using the theorem of Pappus. The cylindrical integral is easy. The spherical one is hard, but Maple gives me the result off by a factor of minus one half for generic values of the two radii, but the correct result for concrete numbers! I cannot figure out what I am doing wrong. Any ideas?
http://www34.homepage.villanova.edu/robert.jantzen/courses/mat2500/handouts/torusvolume.mw

Dear Maplesoft,

I inquired about this problem 4 years ago, but never really was able to fix my problem based on the response at the time. This has to do with plotting a parametrized curve where the parametrization involves the numerical solution of a condition. 

Consider the family of cardioids  
           "r = 1 + c*sin(theta), theta = 0 .. 2*Pi"
 in polar coordinates for 
                         "c = 0 .. 2.5"

In this example we find the polar angle 
                           "theta(c)"
 on the evolving family of cardioids where the slope is 
                              "-1"
 as a function of the shape parameter 
                              "c"
 of this family by a procedure involving fsolve, but then try to plot the parametrized curve 
                  "r(c) = 1 + c*sin(theta(c))"
. No direct plotting method works because of evaluation order problems that I do not understand. The first plot is my desired plot but I used an ugly workaround to get the gray curve. Can you fix the direct method with delayed evaluation or something? 

Maplesoft Response. We don't help with this kind of problem. Ask MaplePrimes. 

This evaluation order problem pops up every time you want to plot a curve determined by numerically solving a condition, yet Maplesoft seems to think this is too sophisticated a problem to respond to. Naively trying to animate such curves always derails, so it reveals a weakness of Maple for users who do not belong to the elite class of Maple experts. I have been using Maple for a quarter century, and have made some pretty intricate animations and plots over the years, but always run up against this problem with animating numerically determined curves. Is there a Maple pro out there who can help?
Since I can't find a way to attach my Maple worksheet, here is the URL:
http://www34.homepage.villanova.edu/robert.jantzen/maple/misc/cardioidfamily.mw