Let $f$ be the function defined by $f(x)=\frac{5}{x}$ . If five subintervals of equal length are used, what is the value of the right Riemann sum approximation for $\int_{2}^{3} \frac{5}{x} d x$ ? Round to the nearest thousandth if necessary. $\newline$ Answer:

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Q. Let

f

be the function defined by

f(x)=\frac{5}{x}

. If five subintervals of equal length are used, what is the value of the right Riemann sum approximation for

\int_{2}^{3} \frac{5}{x} d x

? Round to the nearest thousandth if necessary.

\newline

Answer:

Calculate Subinterval Width: We need to calculate the right Riemann sum for the function $f(x) = \frac{5}{x}$ from $x = 2$ to $x = 3$ using five equal subintervals. First, we find the width of each subinterval. $\newline$ The total interval length is $3 - 2 = 1$ . With five subintervals, each subinterval has a width of $\frac{1}{5}$ .
Determine Right Endpoints: Next, we determine the $x$ -values at the right endpoints of each subinterval. Since we start at $x = 2$ and each subinterval has a width of $\frac{1}{5}$ , the right endpoints are $x = 2.2$ , $x = 2.4$ , $x = 2.6$ , $x = 2.8$ , and $x = 3$ .
Evaluate Function at Endpoints: Now we evaluate the function $f(x) = \frac{5}{x}$ at each of these right endpoints. This gives us the heights of the rectangles for the Riemann sum. $f(2.2) = \frac{5}{2.2}$ $f(2.4) = \frac{5}{2.4}$ $f(2.6) = \frac{5}{2.6}$ $f(2.8) = \frac{5}{2.8}$ $f(3) = \frac{5}{3}$
Calculate Rectangle Areas: We calculate the area of each rectangle by multiplying the height by the width $(1/5)$ . The sum of these areas will give us the right Riemann sum approximation. $\newline$ $\text{Area} = \left(\frac{1}{5}\right) \times \left[f(2.2) + f(2.4) + f(2.6) + f(2.8) + f(3)\right]$ $\newline$ $\text{Area} = \left(\frac{1}{5}\right) \times \left[\left(\frac{5}{2.2}\right) + \left(\frac{5}{2.4}\right) + \left(\frac{5}{2.6}\right) + \left(\frac{5}{2.8}\right) + \left(\frac{5}{3}\right)\right]$
Sum Up Areas: Perform the calculations for each term and sum them up. $\newline$ Area $\approx \frac{1}{5} \times \left[\left(\frac{5}{2.2}\right) + \left(\frac{5}{2.4}\right) + \left(\frac{5}{2.6}\right) + \left(\frac{5}{2.8}\right) + \left(\frac{5}{3}\right)\right]$ $\newline$ Area $\approx \frac{1}{5} \times \left[2.273 + 2.083 + 1.923 + 1.786 + 1.667\right]$ $\newline$ Area $\approx \frac{1}{5} \times 9.732$ $\newline$ Area $\approx 1.9464$