Multivariable Calculus

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Center of Uneven Density Rod

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Assume a thin rod $L$ lies between $0$ and $2$ has density given by:

$$ \begin{equation} \rho(x)= \begin{cases} x^3, \quad \forall x\in[0,1] \\\ 2x-x^2, \quad \forall x\in (1,2] \end{cases} \end{equation} $$

Find the center of mass for $L$.

A

$\cfrac{67}{60}$

B

$\cfrac{67}{55}$

C

$\cfrac{11}{12}$

D

$\cfrac{37}{21}$

E

$\cfrac{12}{11}$