Find the center of mass of a thin plate of constant density covering the given region.
-The region cut from the first quadrant by the circle $x ^ { 2 } + y ^ { 2 } = 49$
A) $\bar { x } = 0 , \bar { y } = \frac { 28 } { 3 \pi }$
B) $\bar { x } = \frac { 7 } { 2 } , \bar { y } = \frac { 7 } { 2 }$
C) $\bar { x } = \frac { 7 } { 3 \pi } , \bar { y } = \frac { 7 } { 3 \pi }$
D) $\bar { x } = \frac { 28 } { 3 \pi } , \bar { y } = \frac { 28 } { 3 \pi }$

Question

Quizplus · Accepted Answer

Since the region is symmetric about both x and y axis, we can assume that the center of mass lies on the origin. We can use double integrals to compute the center of mass.
Let $f(x,y)=1$ be the density function of the plate. We have 
$$\iint_R f(x,y) dA = \iint_R 1dA$$
where R is the region cut from the first quadrant by the circle $x ^ { 2 } + y ^ { 2 } = 49$.
Using polar coordinates, we have 
\begin{align*}
\iint_R 1dA &= \int_{0}^{\frac{\pi}{2}}\int_{0}^{7}  r \, dr \,d\theta\
&=\int_{0}^{\frac{\pi}{2}} \frac{1}{2}(7^2-0^2) d\theta\
&=\frac{49}{2}\left[\theta\right]^{\frac{\pi}{2}}_{0}=\frac{49}{2}\cdot \frac{\pi}{2}
\end{align*}
The total area is $\frac{49}{2}\cdot \frac{\pi}{2}$. Therefore, the x-coordinate of the center of mass is
$$\bar{x} = \frac{\iint_R x f(x,y)dA}{\iint_R f(x,y)dA}=\frac{\iint_R x dA}{Area}.$$
Using polar coordinates, we have 
\begin{align*}
\iint_R x dA &= \int_{0}^{\frac{\pi}{2}}\int_{0}^{7}  r \cos \theta \, r \, dr \,d\theta\
&=\int_{0}^{\frac{\pi}{2}} \cos \theta \, \frac{1}{2}(7^3-0^3) d\theta\
&=\frac{7^3}{2}\left[\sin \theta\right]^{\frac{\pi}{2}}_{0}\
&=\frac{7^3}{2}
\end{align*}
Therefore, the x-coordinate of the center of mass is 
$$\bar{x} = \frac{1}{Area} \iint_R x dA =\frac{1}{\frac{49}{2}\cdot \frac{\pi}{2}} \cdot \frac{7^3}{2}= \frac{28}{3\pi}.$$
Similarly, we can find that the y-coordinate of the center of mass is also $\frac{28}{3\pi}$. Therefore, the center of mass is $(\frac{28}{3\pi},\frac{28}{3\pi})$ and the correct answer choice is $\boxed{\textbf{(D)}}$.

Find the Center of Mass of a Thin Plate of Constant