I need help to calculate the following surface area:
the surface area common to the two cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$ using surface integrals essentially.
My attempt: Let surface area = $S$ and $\hat n = \nabla(x^2 + y^2) = \frac{1}{a}(x \hat i + y \hat j)$ $$S = \int \int ds = \int \int \frac{dy dz}{|\hat n\cdot\hat i|}$$ $$= \int \int \frac{a}{x}dy dz$$ $$= \int \int \frac{a}{\sqrt{a^2 - y^2}}dy dz$$
How do I progress from here on?