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Evaluate the double integral by converting to polar coordinates: $$\int_0^2 \int_0^x \sqrt{x^2 + y^2} \, dy \, dx + \int_2^{2\sqrt{2}} \int_o^{\sqrt{8 - x^2}} \sqrt{x^2 + y^2} \, dy \, dx $$

What I have done: First, I tackled the left hand integral. Using the fact that $x = r \cos\theta$ and $y = r\sin\theta$, I converted the left hand integral to $\int r^2 \, dr \, d\theta$. I drew the graph of the region to get a right triangle. I am having trouble determining the limits of integration for $r$ and $\theta$

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  • $\begingroup$ @E.H.E: I don't think there is. $\endgroup$
    – joriki
    Oct 17, 2015 at 20:35

1 Answer 1

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The two regions together form a sector of a circle with radius $\sqrt8$ centred at the origin, so the corresponding integral in polar coordinates is

$$ \int_0^{\frac\pi4} \, \mathrm d\phi\int_0^\sqrt8r \, \mathrm dr \, r\;. $$

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  • $\begingroup$ Wolfram Alpha also gave me ~5.91 which I obtained with this method $\checkmark$ $\endgroup$
    – WAS
    Oct 18, 2015 at 16:54

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