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# Example of calculating a surface integral part 3

Example of calculating a surface integral part 3. Created by Sal Khan.

## Want to join the conversation?

• the surface area of the torus is the product of the circumferences of the two parameter circles and the volume is the area of the smaller times the circumference of the larger.A very satisfying result. Can this be generalized?
• If a bounded area is intersected by a continuous path through the centroid at a right angle and the area is moved along the parh so that it remains perpendicular, the volume generated is equal to the area timis the length of the path and its lateral area is the product of its circumference times the length of the path. I haven't proned this but I believe it is true.
• I don't know if I'm missing the point but I'm kind of playing devils advocate: Why couldn't you just snip the toroid along the cross section and then stretch it into a cylinder and then snip the cylinder along the length of the cylinder to make it a rectangle? Then you could use a single integral or solve it geometrically.
• If you snip the toroid along the cross section, it won't become a perfect cylinder. It will still have some intrinsic geometry, because the inside of the torus would be longer than the outside.
• If the torus was in some vector field so that the function was not 1 would there be additional steps involved, besides just multiplying by the function, in calculating the integral?
• If the function was a function of (x,y,z) you would have to change it to the parametrisation used and then just multiply, so no extra steps although the "hairiness" would most likely increase.
• It makes perfect sense when you read this:

"The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by its geometric centroid."

https://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem#The_first_theorem

Not bad for someone who existed c. 290 AD. It's really really simple though, it's amazing.
• How do we know that we should use "t" parameter the outside of the double integral, is there a specific rule?
• It's up to you. If you integrate with respect to s first, then you can do with t later or use it 'outside' of the double integral.
(1 vote)
• I wanted to ask why at about b + acoss didn't change. Why it didn't get raised to the 2nd power like the rest of the factors of the vectors. But then I thought maybe it did, but got cancelled out with the 1/2 power over the entire thing. I don't know if I'm right for sure though so could someone tell me for sure? Thanks in advance.
• You are correct, he just simplified it immediately!
• said surface area of the torus is 4*pi^2*a*b..
what if b=0 and a=r, it turns out 4*pi^2*r*0=0, but actually it is surface of sphere that is 4*pi*r^2..
I think something missing from the base..

(1 vote)
• This was a parameterization of a torus, not a sphere, they are topologically different. So why would the surface area of a sphere be the special case of a torus?
• is it possible to say that the surface area of the torus is the product of the circumference of the smaller, inner circle with the larger,outer circle?
(1 vote)
• Indeed that is one way to view it. You can also view the volume of a torus as the product of the area of the smaller circle with the circumference of the outer circle.
http://en.wikipedia.org/wiki/Pappus%27s_centroid_theorem
• What would happen if the inner integrand, integrated with respect to s, had terms involving both s and t, rather than just s? Would we treat any terms involving t as constants? (I'm not sure if 'partial integration' is a legitimate term, but that's how I think about the inverse of a partial derivative)
(1 vote)
• Yes, the t would be treated as a constant.