Thread: Surface area and Volume

A Circle C is given by the equation x^2 +(y-1)^2 = 1. It is rotated about the x-axis to give a surface of revolution called a torus.

a) Find the total surface area of the torus.

(b) Find the volume of the torus and verify that it is 2pi times the area of C.

Have you covered the theorems of Guldin?

No, what we have covered are that :

the surface area is the integral of 2pi y (1+(dy/dx)^2)^1/2 dx, and the volume is the integral of pi y^2 dx. I have used these equations, integrated from -1 to 1 and got 2pi(2+pi) for the area, and 10pi/3 +pi^2. But I think I'm doing something wrong. Can you help me?

Better work in parametric equations . Then, the area is . Let's see what do you obtain.

Originally Posted by FernandoRevilla

Better work in parametric equations . Then, the area is . Let's see what do you obtain.

When I integrated and used the limits 0 and 2pi, I got an answer of 4pi^2, but when I changed to 0 and pi, I got 2pi(pi+2) which is the same answer as when I worked it out using the cartesian equation rather than the parametric. Which one is correct?

And what about the volume? I worked it out using parametric as well but got an answer of 2pi^2. The question says that it should be 2pi times the answer of the area...

Can you help please?

Thanks

Originally Posted by perla

When I integrated and used the limits 0 and 2pi, I got an answer of 4pi^2

Right.

And what about the volume? I worked it out using parametric as well but got an answer of 2pi^2. The question says that it should be 2pi times the answer of the area...

Right, the volume is