Solutions - 0070-01

(1) Recall the formula for the average value of $f$ over $a\le x\le b$:

$$f_{\mathrm{avg}}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$

Here $a=0$ and $b=180$:

$$\gg \gg \frac{1}{180}{\int }_0^{180}f(x)dx$$

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(3) Use $\mathrm{\Delta }x=15$ in Simpson’s Rule:

\begin{align*} \int_{0}^{180}f(x)dx \quad \approx\quad S_{12} &\quad=\quad \frac{15}{3}\Big(21+4(21.3)+2(21.5)+\cdots+4(21.3)+21.2)\Big) \end{align*} ParseError: Invalid delimiter 'undefined' after '\Big'

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(4) Plug into average value formula:

$$f_{\mathrm{avg}}\approx \frac{S_{12}}{180}\gg \gg \approx {21.2}^{\circ }$$