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Chapter 5: Problem 17

Use the figures to calculate the left and right Riemann sums for \(f\) on the given interval and for the given value of \(n\). $$f(x)=x+1 \text { on }[1,6] ; n=5$$

Short Answer

Expert verified
Question: Calculate the left and right Riemann sums for the function \(f(x) = x + 1\) on the interval \([1, 6]\) with \(n = 5\) equal subintervals. Answer: The left Riemann sum is 20, and the right Riemann sum is 25.

Step by step solution

01

Find the width of each subinterval

To determine the width of each subinterval, we'll divide the length of the entire interval by the number of equally sized subintervals (n): $$\Delta x = \frac{b - a}{n} = \frac{6 - 1}{5} = \frac{5}{5} = 1$$ The width of each subinterval, \(\Delta x\), is 1.
02

Determine the left and right endpoints of each subinterval

To find the left and right endpoints of each subinterval, we start from the left endpoint of the entire interval, \(a = 1\), and add multiples of the subinterval width, \(\Delta x = 1\), as follows: Left Endpoints: \(x_0 = 1, x_1 = 2, x_2 = 3, x_3 = 4, x_4 = 5\) Right Endpoints: \(x_1 = 2, x_2 = 3, x_3 = 4, x_4 = 5, x_5 = 6\)
03

Evaluate the function at the left and right endpoints

Now, we'll evaluate the function, \(f(x) = x + 1\), at each of the left and right endpoints: Left endpoints: \(f(x_0) = 2, f(x_1) = 3, f(x_2) = 4, f(x_3) = 5, f(x_4) = 6\) Right endpoints: \(f(x_1) = 3, f(x_2) = 4, f(x_3) = 5, f(x_4) = 6, f(x_5) = 7\)
04

Calculate the left and right Riemann sums

Finally, we'll calculate the left and right Riemann sums by multiplying the function values from the previous step by the subinterval width, \(\Delta x = 1\), and then summing them up: Left Riemann Sum: $$L_n(f)=\sum_{i=0}^{4}f(x_i)\Delta x= (2+3+4+5+6)(1)=20$$ Right Riemann Sum: $$R_n(f)=\sum_{i=1}^{5}f(x_i)\Delta x= (3+4+5+6+7)(1)=25$$ So the left Riemann sum is 20, and the right Riemann sum is 25.

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