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Find the indicated Midpoint Rule approximation to the following integral. 13 |3x dx using n= 1, 2, and 4 subintervals

Find the indicated Midpoint Rule approximation to the following integral. 13 |3x dx using n= 1, 2, and 4 subintervals

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Find the Indicated Midpoint Rule Approximation to the Following Integral** Evaluate the integral \( \int_{5}^{13} 3x^2 \, dx \) using the Midpoint Rule with \( n = 1, 2, \) and \( 4 \) subintervals. --- ### Midpoint Rule Approximations 1. **For \( n = 1 \) subinterval:** - The Midpoint Rule approximation of \( \int_{5}^{13} 3x^2 \, dx \) is \(\_\_\_ \). - *(Round to three decimal places as needed.)* 2. **For \( n = 2 \) subintervals:** - The Midpoint Rule approximation of \( \int_{5}^{13} 3x^2 \, dx \) is \(\_\_\_ \). - *(Round to three decimal places as needed.)* 3. **For \( n = 4 \) subintervals:** - The Midpoint Rule approximation of \( \int_{5}^{13} 3x^2 \, dx \) is \(\_\_\_ \). - *(Round to three decimal places as needed.)* --- ### Explanation **Midpoint Rule**: This numerical method is used to approximate the definite integral of a function. It involves dividing the interval into equal subintervals, calculating the midpoint of each subinterval, and evaluating the function at these midpoints. The sum of these values, multiplied by the width of the subintervals, gives the approximation of the integral.
Transcribed Image Text:**Find the Indicated Midpoint Rule Approximation to the Following Integral** Evaluate the integral \( \int_{5}^{13} 3x^2 \, dx \) using the Midpoint Rule with \( n = 1, 2, \) and \( 4 \) subintervals. --- ### Midpoint Rule Approximations 1. **For \( n = 1 \) subinterval:** - The Midpoint Rule approximation of \( \int_{5}^{13} 3x^2 \, dx \) is \(\_\_\_ \). - *(Round to three decimal places as needed.)* 2. **For \( n = 2 \) subintervals:** - The Midpoint Rule approximation of \( \int_{5}^{13} 3x^2 \, dx \) is \(\_\_\_ \). - *(Round to three decimal places as needed.)* 3. **For \( n = 4 \) subintervals:** - The Midpoint Rule approximation of \( \int_{5}^{13} 3x^2 \, dx \) is \(\_\_\_ \). - *(Round to three decimal places as needed.)* --- ### Explanation **Midpoint Rule**: This numerical method is used to approximate the definite integral of a function. It involves dividing the interval into equal subintervals, calculating the midpoint of each subinterval, and evaluating the function at these midpoints. The sum of these values, multiplied by the width of the subintervals, gives the approximation of the integral.
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