A bizarre, mind-bending discrepancy (volumes of revolution)

[rogerstein]

I noticed an exceedingly puzzling discrepancy that I cannot resolve. It involves a solid of revolution 1)generated in the normal way and 2)generated parametrically. Obviously, when done properly the two answers should be the same, as it has been for me in a number of instances. However, in the case at hand, there's a mysterious disparity. The equations are y=4/t and x=4t. Expressed in the usual way, y=16/x. Now, let's do it with y=16/x first. The solid of revolution is around the y-axis (yes, the y-axis, not the x--incidentally, I experimentally rotated it around the x-axis and calculated both ways, and, appropriately, the two answers were the same) and the boundaries are x=4, x=16, the x-axis, and y=16/x. Using cylindrical shells, the integral, deliberately not simplified, is:
Integral {4,16} 2*pi*x*(16/x) The antiderivative is 32*pi*x and the answer is 384*pi. I also did it, quite laboriously, using cross-sectional areas, and the result was identical. Now comes the strangeness! Doing it parametrically, y=4/t and x=4t. First, we convert the boundaries. x=4 becomes t=1 and x=16 becomes t=4. Now to the integral, where I will treat y=4/t as a normal equation and x=4t as the one to be differentiated and multiplied with the first, as is normally done when doing things parametrically. Using cylindrical shells, with new limits of integration, again purposely not simplifying, the integral is:
Integral {1,4} 2*pi*t*(4/t)*4 The antiderivative is 32*pi*t and the answer is 96*pi. (Instead of 384*pi) Good grief, I've gone terribly wrong, but how, where, why? Regarding the last integral, the 4 at the right is the derivative of x=4t. And please note this: when all the data was the same but I rotated it around the x-axis, both the normal version and the parametric version yielded the same answer, 48*pi. When doing the parametric version in the 48*pi case, I did it exactly as I did it in the example causing the discrepancy, except I used the cross-sectional areas technique. This all constitutes an extremely disturbing and really quite mind-boggling state of affairs and I'm hoping someone comes galloping to my rescue.

 

[skeeter]

Re: A bizarre, mind-bending discrepancy

for cylindrical shells about the y-axis, the basic equation in rectangular coordinates is ...

\(\displaystyle V = 2\pi \int_a^b x \cdot y \cdot dx\)

\(\displaystyle x = 4t\)
\(\displaystyle y = \frac{4}{t}\)
\(\displaystyle dx = 4 \, dt\)

\(\displaystyle V = 2\pi \int_1^4 4t \cdot \frac{4}{t} \cdot 4 dt = 2\pi[64t]_1^4 = \2pi[256 - 64] = 384\pi\)

 

[rogerstein]

But Skeeter, are you saying that for all other parametric procedures (areas under a curve, volumes of revolution calculated using cross-sectional areas) you simply use the y equation and the derivative of x, but for this single procedure, you use the x and the y as well as the derivative of x? Why should there be that kind of inconsistency?

 

[skeeter]

I fail to see your point ... my derivation for volume by cylindrical shells is rather clear.

your attempt at a calculation ...

Integral {1,4} 2*pi*t*(4/t)*4

is lacking. it should be ... Integral {1,4} 2*pi*4t*(4/t)*4 dt

you missed a 4 since x = 4t.

 

[rogerstein]

Here's my point, Skeeter: Let's take a simple example. We want to calculate the area under a curve, two ways--normally, and parametrically. So here are the equations: y=t^3 +2 and x=t^2. You want the area from t=0 to t=3. First, let's translate this so that it's entirely in terms of x and y. x=t^2, so t=sqrt x. So we substitute back into y=t^3 + 2 and get y= (sqrt x)^3 + 2. We reset the limits of integration, t=0 is x=0 and t=3 becomes x=9. So now we integrate:
Integral {0,9} (sqrt x)^3 + 2 The antiderivative is .4 x^2.5 + 2x, from 0 to 9. The answer is 115.2.
Now let's do it parametrically: y=t^3 + 2 and x=t^2, from t=0 to t=3.
Integral {0,3} (t^3 + 2)*(2t) The antiderivative is .4 *t^5 +2* t^2, from 0 to 3. The answer is 115.2. The same answer as above, properly so. But here's the point: the second integral (the parametric one), has only 2 terms. The first is the y equation, the second is the derivative of the x equation. And that combination produces the correct answer. Similarly, when doing solids of revolution, when using cross-sectional areas as the technique, I did exactly the same thing, and got matching answers,i.e. I used the y equation and the derivative of the x, and did not include the x equation. So, I'm wondering why when doing solids of revolution by cylindrical shells, there's this huge change, and you have to include the x equation as well as the x derivative. That's the puzzle. When I submitted my question I knew that the method I used wasn't correct, but I didn't (and still don't) understand why you have one set of procedures for areas and solids of revolution done by cross-sectional areas and a different parametric procedure (where you suddenly include the x equation too) when you do cylindrical shells. That sort of inconsistency is not like anything I've ever seen in mathematics.

 

[skeeter]

what makes you think that the set up for finding volume using cylindrical shells is the same as finding area?

go have a talk with your prof.