Solutions - 0070-03

Note: you can also start from $y_0=4.5$ and $y_8=4.0$. This gives a different answer, $\approx 1.3$.

(1) Set up integration:

Set $(x,y)=(\mathrm{0,0})$ at the left upper corner, with $+x$ extending to the right, $+y$ extending downwards. Then:

$$A={\int }_0^{20}f(x)dx$$

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(2) Create table of values:

$i$	$x_i$	$f(x_i)$
$0$	$0$	$0$
$1$	$2$	$4.5$
$2$	$4$	$3.3$
$3$	$6$	$3.3$
$4$	$8$	$5$
$5$	$10$	$6$
$6$	$12$	$5$
$7$	$14$	$4$
$8$	$16$	$5$
$9$	$18$	$4$
$10$	$20$	$0$

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(3) Recall Simpson’s Rule formula:

$$S_n=\frac{1}{3}\mathrm{\Delta }x(y_0+4y_1+2y_2+\cdots +2y_{n-2}+4y_{n-1}+y_n)$$

Here $y_i=f(x_i)$ and $\mathrm{\Delta }x=2$.

Thus:

$$\begin{array}{cc}A\approx S_{10}& =\frac{1}{3}\cdot 2\cdot (0+4\cdot 4.5+2\cdot 3.3+4\cdot 3.3+\cdots +4\cdot 4+0)\\ & \\ & \approx 82.53\end{array}$$

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(4) Compute cubic yards from known surface area:

Mulch is $0.5\mathrm{ft}$ deep, so the volume is:

$$\begin{array}{cc}82.53{\mathrm{ft}}^2\times 0.5\mathrm{ft}& \gg \gg 41.265{\mathrm{ft}}^3\\ & \\ & \gg \gg 1.53{\mathrm{yd}}^3\end{array}$$