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Chapter 5: Problem 18

Use the figures to calculate the left and right Riemann sums for \(f\) on the given interval and for the given value of \(n\). $$f(x)=\frac{1}{x} \text { on }[1,5] ; n=4$$

Short Answer

Expert verified
Question: Calculate the left and right Riemann sums for the function \(f(x) = \frac{1}{x}\) on the interval \([1, 5]\) with \(n = 4\) rectangles. Answer: The left Riemann sum is approximately \(2.08\), and the right Riemann sum is approximately \(1.28\).

Step by step solution

01

Determine the width of each rectangle

First, we need to find the width of each rectangle. To do that, we can subtract the starting point of the interval from the ending point and divide it by the number of rectangles (\(n\)). In this case, our interval is \([1, 5]\) and \(n = 4\). So the width of each rectangle is: $$\Delta x = \frac{5 - 1}{4} = 1$$
02

Determine the left endpoints of each subinterval

To find the left Riemann sum, we need to determine the left endpoints of each subinterval. We start from the left endpoint of the interval (\(1\)) and move to the right by a width of \(\Delta x = 1\). The left endpoints for each subinterval are \(x_1 = 1, x_2 = 2, x_3 = 3,\) and \(x_4 = 4\).
03

Calculate the left Riemann sum

Now, we evaluate the function \(f(x)\) at each left endpoint and multiply the result by the width (\(\Delta x\)) of the rectangle, and then sum all the products: $$L_n = \sum_{i=1}^4{f(x_i)\Delta x}$$ $$= (\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4})(1) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = 25/12 \approx 2.08$$
04

Determine the right endpoints of each subinterval

Now, we need to find the right Riemann sum, for which we need to find the right endpoints of each subinterval. We start from the right endpoint of the first subinterval (\(x_1 = 2\)), and move to the right by a width of \(\Delta x = 1\). The right endpoints for each subinterval are \(x_1' = 2, x_2' = 3, x_3' = 4\), and \(x_4' = 5\).
05

Calculate the right Riemann sum

We evaluate the function \(f(x)\) at each right endpoint and multiply the result by the width (\(\Delta x\)) of the rectangle, and then sum all the products: $$R_n = \sum_{i=1}^4{f(x_i')\Delta x}$$ $$= (\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5})(1) = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = 77/60 \approx 1.28$$ So the left Riemann sum is approximately \(2.08\), and the right Riemann sum is approximately \(1.28\).

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