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How do I evaluate this double integral

How do I evaluate this double integral-example-1

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Convert to polar coordinates with


x = r \cos(\theta)


y = r \sin(\theta)

so that
x^2 + y^2 = r^2, and the Jacobian determinant for this change of variables is


dx\,dy = r \, dr \, d\theta

D is the disk centered at the origin with radius 2; in polar coordinates, this is the set


D = \left\{(r, \theta) \mid 0\le\theta\le2\pi \text{ and } 0 \le r \le 2\right\}

Then the integral is


\displaystyle \iint_D (x + y + 10) \, dx \, dy = \int_0^(2\pi) \int_0^2 (r \cos(\theta) + r \sin(\theta) + 10) r \, dr \, d\theta


\displaystyle = \int_0^(2\pi) \int_0^2 (r^2 \cos(\theta) + r^2 \sin(\theta) + 10r) \, dr \, d\theta


\displaystyle = \int_0^(2\pi) \left(\frac83 (\cos(\theta) + \sin(\theta)) + 20\right) \, d\theta


\displaystyle = 20 \int_0^(2\pi) d\theta = \boxed{40\pi}

(since cos and sin are 2π-periodic)

User Amillerrhodes
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