R2D2 Volume Equation Step by Step

I saw an R2D2 volume equation on imgur the other day.  I did some math and it came wrong when no substitutions are allowed.   But we can achieve "R2D2" it if we allow substitution

Img src: http://imgur.com/ITRq8OW

Similarly: http://www.starwars-inspired.com/post/13774331797/r2d2-equation
Without any substitutions... it is not possible to get the volume indicated by the Yoda cartoon.  But it is possible to achieve it with substitutions.

Img src: http://shirtoid.com/130230/volume/

This is the actual formula that you can achieve without using substitution.  It is not quite the correct form.

 Now, we want to achieve R2D2 in the form shown on the first image.  But how?  Let's do the work.

Volumes of shapes we need.

Img src: http://www.math.fsu.edu/~wooland/hm2ed/Part3Module9/Sol4/Sol4.html

Img src: http://cs.uwec.edu/~buipj/teaching/cs.170.s15/lab_01.html

 We'll say that V1 is the volume of a sphere, and V2 is the volume of the cylinder.  The two volumes will give us the volume of R2D2, but remember that the volume of the sphere has to be cut in half, so V1/2

Known: D = 2R  , R = D/2

Here are substitutions that we'll need:  H = (5/3)D   ,  D = 2Pi    (you could also say that R = Pi too)

Volume of body = V1/2 + V2         
= (4/3)PiRRR(1/2) + PiRRH            -- Place the equations, and divide V1 by 1/2.  For web I used RRR instead of R^3, etc.
= (2/3)PiRRR + PiRRH                  -- The result.
= (2/3)PiRRR + PiRR(5/3)D            -- Substitute H
= (2/3)PiRR(D/2) + PiRR(5/3)D        -- Substitute one of the Rs
= (1/3)PiRRD + PiRR(5/3)D             -- Simplify
= PiRRD([1/3]+[5/3])                       -- Factor out PiRRD
= PiRRD(6/3)                                 -- Add the fractions
= PiRRD2                                      --  The result  (this is normally where one would stop)
= 2PiRRD                                      -- Rearrange
= DRRD                                        -- Substitute 2Pi with D
= RRDD                                        -- Rearrange
= (R^2)(D^2)                                   -- aka R2D2

Now, the substitution of H means that we have given it some actual numbers.  What does the shape of look like with the given value.

 Not too bad!

There may be something wrong though... a volume is something to the 3rd.  The (R^2)(D^2) would give something to the 4th. Oh well... you can't get them all.

Note: There could be errors, so always double check!