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

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Q. Let

f

be the function defined by

f(x)=2^{x}

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

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

? Round to the nearest thousandth if necessary.

\newline

Answer:

Determine Width of Subinterval: Determine the width of each subinterval. The interval from $3$ to $5$ has a length of $5 - 3 = 2$ . Since we are using five subintervals of equal length, each subinterval will have a width of $\frac{2}{5}$ .
Identify Left Endpoints: Identify the $x$ -values for the left endpoints of each subinterval. The left endpoints will be at $x = 3$ , $3 + \frac{2}{5}$ , $3 + 2\left(\frac{2}{5}\right)$ , $3 + 3\left(\frac{2}{5}\right)$ , and $3 + 4\left(\frac{2}{5}\right)$ . These values are $x = 3$ , $3.4$ , $3.8$ , $4.2$ , and $x = 3$ $0$ .
Evaluate Function Values: Evaluate the function $f(x) = 2^x$ at each left endpoint. $f(3) = 2^3 = 8$ $f(3.4) = 2^{3.4} \approx 10.556$ $f(3.8) = 2^{3.8} \approx 13.964$ $f(4.2) = 2^{4.2} \approx 18.475$ $f(4.6) = 2^{4.6} \approx 24.455$
Calculate Rectangle Areas: Multiply each function value by the width of the subintervals to find the area of each rectangle. $\newline$ The width of each subinterval is $\frac{2}{5}$ , so we multiply each function value by $\frac{2}{5}$ : $\newline$ $\text{Area}_1 = f(3) \times \left(\frac{2}{5}\right) = 8 \times \left(\frac{2}{5}\right) = 3.2$ $\newline$ $\text{Area}_2 = f(3.4) \times \left(\frac{2}{5}\right) \approx 10.556 \times \left(\frac{2}{5}\right) \approx 4.222$ $\newline$ $\text{Area}_3 = f(3.8) \times \left(\frac{2}{5}\right) \approx 13.964 \times \left(\frac{2}{5}\right) \approx 5.586$ $\newline$ $\text{Area}_4 = f(4.2) \times \left(\frac{2}{5}\right) \approx 18.475 \times \left(\frac{2}{5}\right) \approx 7.39$ $\newline$ $\text{Area}_5 = f(4.6) \times \left(\frac{2}{5}\right) \approx 24.455 \times \left(\frac{2}{5}\right) \approx 9.782$
Find Riemann Sum: Add the areas of the rectangles to find the left Riemann sum approximation. $\newline$ Left Riemann Sum $= \text{Area}_1 + \text{Area}_2 + \text{Area}_3 + \text{Area}_4 + \text{Area}_5$ $\newline$ $\approx 3.2 + 4.222 + 5.586 + 7.39 + 9.782$ $\newline$ $\approx 30.18$