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Multivariable Calculus

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Centroid of 2D Polar Region

MVCALC-1ROGJK

Find the centroid (center of mass) of the unit density region $R$, where $R$ is given in polar coordinates:

$$R=\{(r,\theta): r\leq a, r\leq a(1+\cos \theta)\}, \quad a>0$$

A

$(0,\cfrac{(15\pi-32)a}{30\pi-48})$

B

$(0,\cfrac{(5\pi - 8)a}{10\pi})$

C

$(\cfrac{(15\pi-32)a}{30\pi-48},0)$

D

$(\cfrac{(5\pi - 8)a}{10\pi}, 0)$

E

None of the above