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Question

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

Solution

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

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

(The same formula is used for the $y$ coordinate, you just need to substitute $x$ for $y$).

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

\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 $\boxed{(0,0)}$ and $\boxed{(2,4)}$. We will need this in the integration to determine the boundaries.

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