MATH200 December 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 • Q5 (a) • Q5 (b) • Q6 • Q7 • Q8 (a) • Q8 (b) i • Q8 (b) ii • Q8 (b) iii •
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If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Recall that the yvalue of the center of mass is given by
$\displaystyle {\overline {y}}={\frac {\iint _{R}y\rho (x,y)\;{\text{d}}A}{\iint _{R}\rho (x,y)\;{\text{d}}A}}$

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Solution

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Since the region of integration is a triangle with vertices $\displaystyle (1,0),\ (2,0),\ (0,2)$ the region is bounded by the function $\displaystyle y=2x$ and $\displaystyle y=22x$, see sketch
We want to integrate with respect to $\displaystyle x$ first, hence, solving for x, the integration boundaries become
 $\displaystyle {\begin{aligned}\textstyle {\frac {1}{2}}(2y)&\leq x\leq 2y\\0&\leq y\leq 2\end{aligned}}$
and the region is $\displaystyle R=\{(x,y)\in \mathbb {R} ^{2}\ 1{\frac {y}{2}}\leq x\leq 2y,\ 0\leq y\leq 2\}$
To calculate $\displaystyle {\overline {y}}={\frac {\iint _{R}y\rho (x,y)\;{\text{d}}A}{\iint _{R}\rho (x,y)\;{\text{d}}A}}$
which is the $\displaystyle y$ coordinate of the center of mass,
we calculate at first the integral in the numerator.
$\displaystyle {\begin{aligned}\iint _{R}y\rho (x,y)\;{\text{d}}A&=\int _{0}^{2}\int _{1{\frac {y}{2}}}^{2y}y^{3}\;{\text{d}}x{\text{d}}y\\&=\int _{0}^{2}y^{3}(2y1+{\frac {y}{2}})\;{\text{d}}y\\&=\int _{0}^{2}(y^{3}{\frac {y^{4}}{2}})\;{\text{d}}y\\&={\frac {2^{4}}{4}}{\frac {2^{5}}{10}}={\frac {4}{5}}\end{aligned}}$
The integral in the denominator is very similar
$\displaystyle {\begin{aligned}\iint _{R}\rho (x,y)\;{\text{d}}A=\int _{0}^{2}\int _{1{\frac {y}{2}}}^{2y}y^{2}\;{\text{d}}x{\text{d}}y=\int _{0}^{2}(y^{2}{\frac {y^{3}}{2}})\;{\text{d}}y={\frac {2}{3}}\end{aligned}}$
Hence
$\displaystyle {\overline {y}}={\frac {\frac {4}{5}}{\frac {2}{3}}}={\frac {6}{5}}$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Multiple integral, MER Tag Multivariable calculus, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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