(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the area of the portion of the cylinder x^2 + y^2 = 2x that lies inside the hemisphere x^2 + y^2 + z^2 = 4, z [tex]\geq[/tex] 0. Hint: Project onto the xz-plane.

2. Relevant equations

I want to use the formula for surface area

[tex]\int\int\frac{|\nabla f|}{|\nabla f\bullet\vec{p}|}dA[/tex]

3. The attempt at a solution

I'm going to consider only the surface in the first octant (for reasons of symmetry). I get

[tex]\frac{|\nabla f|}{|\nabla f\bullet\vec{p}|} = \frac{1}{y}[/tex]

hence:

[tex]\int\int\frac{|\nabla f|}{|\nabla f\bullet\vec{p}|}dA = \int\int\frac{1}{\sqrt{2x-x^2}}dzdx[/tex]

and using

sqrt(4-x^2) and 0 as limits of integration for z

and 2 and 0 as limits of integration for x, I get

[tex]\int\sqrt{\frac{2+x}{x}}dx[/tex]

(with 2 and 0 as limits of integration for x)

The problem is that this integral doesn't evaluate to 4, which I know is the correct answer (I do get this result by evaluating the integral

[tex]\int h ds[/tex]

where h is the altitude of the cylinder and ds is the element of arc length on the circle x^2 + y^2 = 2x in the xy-plane)

Could you please tell me where I'm going wrong?

Many thanks in advance!

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# Homework Help: Hemisphere cut by cylinder

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