Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

A Steinmetz solid is the intersection of two cylinders. My question is how to find the surface area of one with both cylinders of radius 1, without parameterizing the solid.

I parameterized it; that is, used $r(u,v)=\langle \sqrt{1-v^2}, u \sqrt{1-v^2}, v \rangle$, $u,v \in [-1,1]$, and integrated the magnitude of the cross product of the partials: $$\int_{-1}^1 \int_{-1}^1 |r_u \times r_v| dR$$

How can I make the solid into a function $z=f(x,y)$ rather than parameterizing it?

share|cite|improve this question
up vote 4 down vote accepted

Taking the axes of the cylinders along $x$ and $y$, you have $z=\sqrt{1-(\max{(|x|,|y|)})^2}$. So you have to integrate $$\int_{-1}^1 \int_{-x}^x f(x) dx dy + \int_{-1}^1 \int_{-y}^y f(y) dx dy$$ where $f$ is the area element. The first integral gets x>y and the second gets y>x. By symmetry they are the same.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.