Solutions - 0110-04

(1) Simpson’s Rule formula:

$${\int }_a^{b}f(x)dx\approx S_n=\frac{\mathrm{\Delta }x}{3}(f(x_0)+4f(x_1)+2f(x_2)+\cdots +f(x_n))$$

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(2) Simpson’s for total mass $m$:

$$\mathrm{\Delta }x=\frac{b-a}{n}\gg \gg \frac{8-2}{6}\gg \gg 1$$

Therefore:

$$\begin{array}{c}{S}_6\gg \gg \frac{1}{3}(0+4(1)+2(3)+4(3)+2(2)+4(3)+0)\\ \\ \gg \gg \frac{38}{3}\end{array}$$

So:

$$m\gg \gg \rho {\int }_2^{8}f(x)dx\gg \gg \approx \frac{38}{3}\rho$$

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(3) Simpson’s for moment to $x$-axis:

Integral formula:

$$M_x={\int }_2^8\frac{\rho }{2}f{(x)}^{2}dx$$

Approximate with $S_6$:

$$\begin{array}{c}{M}_x\approx S_6\\ \\ \gg \gg \frac{1}{3}\rho (\frac{1}{2}{0}^2+4\frac{1}{2}\left(1^2\right)+2\frac{1}{2}\left(3^2\right)+4\frac{1}{2}\left(3^2\right)+2\frac{1}{2}\left(2^2\right)+4\frac{1}{2}\left(3^2\right)+\frac{1}{2}{0}^2)\\ \\ \gg \gg \frac{102\rho }{6}\end{array}$$

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(4) Simpson’s for moment to $y$-axis:

Integral formula:

$$M_y={\int }_2^8\rho xf(x)dx$$

Approximate with $S_6$:

$$\begin{array}{c}{M}_y\approx S_6\\ \\ \gg \gg \frac{1}{3}\rho (1(2)(0)+4(3)(1)+2(4)(3)+4(5)(3)+2(6)(2)+4(7)(3)+1(8)(0))\\ \\ \gg \gg 68\rho \end{array}$$

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(5) Compute CoM:

$$\begin{array}{cc}\bar{x}& =\frac{M_y}{m}\gg \gg \frac{68}{38/3}\gg \gg \frac{102}{19}\approx 5.37\\ & \\ \bar{y}& =\frac{M_x}{m}\gg \gg \frac{17}{38/3}\gg \gg \frac{51}{38}\approx 1.34\end{array}$$