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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?

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up vote 3 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.

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