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Find the center of mass of a plate that is shaped like the region between y=x2y = x^2 and y=2xy = 2x, where the density varies as 1+x+y.1 + x + y.


Step 1
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To calculate this we use the formula for the $\textbf{Continuous center of mass in R2R^2}$, which is given as

Center of mass x=total moment with respect to y-axistotal mass=Dxδ(x,y)dADδ(x,y)dA\text{Center of mass }\overline{x}=\dfrac{\text{total moment with respect to $y$-axis}}{\text{total mass}}=\dfrac{\int\int_D x\delta(x,y)\,dA}{\int\int_D \delta(x,y)\,dA}

(The same formula is used for the yy coordinate, you just need to substitute xx for yy).

First, find the points of intersection between y=x2y=x^2 and y=2xy=2x:

(x,y)y=x2 and (x,y)y=2x    x2=2x    x(x2)=0    x=0 and x=2\begin{align*} (x,y)\in y=x^2\text{ and } (x,y)\in y=2x &\iff\\ x^2=2x&\iff\\ x(x-2)=0&\iff\\ x=0 \text{ and }x=2& \end{align*}

Therefore, the points of intersection are (0,0)\boxed{(0,0)} and (2,4)\boxed{(2,4)}. We will need this in the integration to determine the boundaries.

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