1. ## no parametric integration

I have a question because I was wondering about something.

Why is it that calculus texts do not cover solids of revolution with respect to parametrics?. Yes, they cover surface area and arc length in parametric, but never mention volume in parametric form. I have seen several problems arise on occasion, but I have never seen a calc text mention this. It's even rough to find something doing a search on the net.

2. Originally Posted by galactus
I have a question because I was wondering about something.

Why is it that calculus texts do not cover solids of revolution with respect to parametrics?. Yes, they cover surface area and arc length in parametric, but never mention volume in parametric form. I have seen several problems arise on occasion, but I have never seen a calc text mention this. It's even rough to find something doing a search on the net.
Mine does

"Calculus: Alternate Fifth edition: Larson/Hostetler/Edwards"

Chapter 12 section two pg. 692

If a smooth curve C given by $x=f(t)$ and $y=g(t)$ does not cross itself on an interval $a\leq{t}\leq{b}$, then the area S of the surface of revolution fomred by C about the coordinate axes is given by the following:

$S=2\pi\int_a^{b}g(t)\sqrt{\bigg(\frac{dx}{dt}\bigg )^2+\bigg(\frac{dy}{dt}\bigg)^2}dt$ Revolution about x-axis g(t)>0

and $S=2\pi\int_{a}^{b}f(t)\sqrt{\bigg(\frac{dx}{dt}\bi gg)^2+\bigg(\frac{dy}{dt}\bigg)^2}dt$ Revolution about y-axis f(t)>0

This is the best calculus book I have ever used

3. As I mentioned, NOT surface area. They all touch on that. I mean volume in terms of parametrics.

Yes, I have that calc text only the 6th ed. Very good calc book. I like a lot of the applied problems

4. Probably the reason it is not touched on is because volume would be ill-defined in parametrics. All the loopdy-loops and such.

5. I have played around with this in the past and recently and I think I MAY have something interesting.

Something simple as an example. Suppose we have a circle of radius 2 using parametrics. $x=2cos(t), \;\ y=2sin(t)$

Then maybe we can use ${\pi}\int_{a}^{b}[y(t)]^{2}\cdot{|x'(t)|}dt$

This would give us ${\pi}\int_{0}^{\pi}\left[(2sin(t))^{2}\cdot{2sin(t)}\right]dt=\frac{32{\pi}}{3}$

Which is the volume of a sphere of radius 2. What we should have when revolving the circle.

Whatcha think?. Cool?. Not?.

I would say with a simple shape like a circle it is OK. Other more complex forms may not work so well. We may not be getting a volume as we would like.