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Find the surface area of the upper half of the sphere $x^2 + y^2 + z^2 = 1$ that is inside the (infinite-height) cylinder $x^2 + y^2 - y = 0.$ If the surface of the sphere has mass per unit area equal to |x|, find the mass of the above surface area.

the first line of the solution has already got me stuck, it says that the cylinder equation can be rewritten as $x^2 + (y - \frac12)^2 = \frac12^2$

can someone please explain this to me? I'm not entirely sure how they reached this! :S

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  • $\begingroup$ The re-writing of the cylinder equation is just completing the square with respect to $y$. $\endgroup$ – Larry B. Oct 20 '16 at 20:13
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Remember that $\left(y - \frac12\right)^2 = y^2 - y + \left(\frac12\right)^2$. Therefore $$x^2 + \left(y - \frac12\right)^2 = \left(\frac12\right)^2$$ is equivalent to $$x^2 + y^2 - y + \left(\frac12\right)^2 = \left(\frac12\right)^2$$ which is equivalent to $$x^2 + y^2 - y = 0.$$

Since these equations are equivalent, any one of them is just as good as another to use as the equation of the cylinder.

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  • $\begingroup$ this has helped so much, i'm doing a bloody degree and I forget such a basic concept! thank you! $\endgroup$ – Sebastian TG Oct 20 '16 at 22:00

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