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

This isn't actually a homework problem, but rather a class of problems I'm running into as I study for prelims. I'm taking these from Greenspan'sCalculus: An Introduction to Applied Mathematics. This type of problem has come up in the context of both volume and surface area, but I'm hoping figuring out the surface area one will be sufficient to figure out the volume one. Here's an example:

Find the surface area of the sphere [itex]x^2+y^2+z^2 = a^2[/itex] contained within the cylinder [itex]x^2+y^2 = ax.[/itex]

2. Relevant equations

Surface area can be determined using a surface integral, [itex]\iint_S\ dS[/itex], and in this case, [itex]dS = a\sin\theta\ d\theta d\phi[/itex]

3. The attempt at a solution

My original thought was to set the two equations equal to each other to get an expression for z. This results in [itex]z = \pm\sqrt{a^2-ax}[/itex]. I've seen one strategy where you put this into the surface integral equation instead of just dS, though I don't understand why, and even if you did, what would the limits of integration be? Just 0 to 2[itex]\pi[/itex] and 0 to [itex]\pi[/itx]?

Thanks!

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# Homework Help: Finding the surface area of one volume contained within another

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