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If the Midpoint Rule is used on the interval \( [-1,17] \) with \( n=3 \) subintervals, at what \( x \)-coordinates is the integrand evaluated? \( \square \) (Simplify your answer. Use a comma to separate answers as needed.)

Ask by Vargas Greene. in the United States
Mar 10,2025

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Tutor-Verified Answer

Answer

The integrand is evaluated at 2, 8, 14.

Solution

The interval is \([ -1, 17 ]\) with \( n = 3 \) subintervals. 1. First, compute the width of each subinterval: \[ \Delta x = \frac{17 - (-1)}{3} = \frac{18}{3} = 6. \] 2. The subintervals are: - \([ -1, -1 + 6] = [ -1, 5 ]\) - \([ 5, 5 + 6] = [ 5, 11 ]\) - \([ 11, 11 + 6] = [ 11, 17 ]\) 3. The midpoints of these subintervals (where the integrand is evaluated) are calculated as follows: - For \([ -1, 5 ]\): \[ x = \frac{-1 + 5}{2} = \frac{4}{2} = 2. \] - For \([ 5, 11 ]\): \[ x = \frac{5 + 11}{2} = \frac{16}{2} = 8. \] - For \([ 11, 17 ]\): \[ x = \frac{11 + 17}{2} = \frac{28}{2} = 14. \] Thus, the integrand is evaluated at \[ 2, 8, 14. \]

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The Deep Dive

To apply the Midpoint Rule, we first determine the width of each subinterval. The interval \( [-1, 17] \) has a total length of \( 17 - (-1) = 18 \). Dividing this by \( n = 3 \) gives us a subinterval width \( \Delta x = \frac{18}{3} = 6 \). The subintervals are: 1. From \( -1 \) to \( 5 \) 2. From \( 5 \) to \( 11 \) 3. From \( 11 \) to \( 17 \) The midpoints of these subintervals will be where we evaluate the integrand: 1. The midpoint of the first interval \( [-1, 5] \) is \( \frac{-1 + 5}{2} = 2 \). 2. The midpoint of the second interval \( [5, 11] \) is \( \frac{5 + 11}{2} = 8 \). 3. The midpoint of the third interval \( [11, 17] \) is \( \frac{11 + 17}{2} = 14 \). Hence, the \( x \)-coordinates at which the integrand is evaluated are \( 2, 8, 14 \).

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