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SCALC$94\mathrm{.}2.$AE$\mathrm{.}006.$

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EXAMPLE $5$ Use the Midpoint Rule with $n=5$ to approximate the following integral.

${\displaystyle \underset{1}{\overset{5}{}}}\frac{2}{x}dx$

SOLUTION The endpoints of the subintervals are $1,1\mathrm{.}8,2\mathrm{.}6,3\mathrm{.}4,4\mathrm{.}2,$ and $5,$ so the midpoints are $1\mathrm{.}4,2\mathrm{.}2,3,3\mathrm{.}8,$ and The width of the subintervals is $x=\frac{5-1}{5}=$

$$

so the Midpoint Rule gives

Since $f(x)=\frac{2}{x}>0$ for $1x5,$ the integral represents an area, and the approximation given by the Midpoint Rule is the sum of the areas of the rectangles shown in the figure.

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SCALC$94\mathrm{.}2.$AE$\mathrm{.}008.$

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