My Math Forum double integral problem

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 October 31st, 2017, 04:21 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 163 Thanks: 2 double integral problem Find the area of z = x^2 - y^2 that lies inside x^2 + y^2 = a^2. Here is my solution. But I am stuck here because I don't know how to calculate this mess... $\displaystyle x = u, y = v, z = u^2-v^2$ $\displaystyle \int_{-a}^{a} \int_{- \sqrt{a^2-u^2}}^{ \sqrt{a^2-u^2}} \sqrt{4u^2+4v^2+1} ~dv ~du$
 October 31st, 2017, 06:40 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 1,692 Thanks: 860 you've confused yourself using u's and v's but otherwise what you've done is correct. however.... when you see $x^2 + y^2 = a^2$ you should immediately be thinking cylindrical coordinates. here we have $\displaystyle \int_0^{2\pi}\int_0^a~\sqrt{4r^2 + 1}~r~dr~d\theta$ and this is trivial to integrate. Thanks from topsquark and zollen

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