/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Suppose the interval [2,6] is pa... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 6

Suppose the interval [2,6] is partitioned into \(n=4\) subintervals with grid points \(x_{0}=2, x_{1}=3, x_{2}=4, x_{3}=5,\) and \(x_{4}=6\) Write, but do not evaluate, the left, right, and midpoint Riemann sums for \(f(x)=x^{2}\).

Short Answer

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Question: Calculate the left, right, and midpoint Riemann sums of the function \(f(x)=x^2\) over the interval \([2,6]\) with \(n=4\) subintervals and given grid points \(x_0, x_1, x_2, x_3,\) and \(x_4\). Answer: Left Riemann Sum: \(L_4 = [2^2 + 3^2 + 4^2 + 5^2]\) Right Riemann Sum: \(R_4 = [3^2 + 4^2 + 5^2 + 6^2]\) Midpoint Riemann Sum: \(M_4 = [2.5^2 + 3.5^2 + 4.5^2 + 5.5^2]\)

Step by step solution

01

Understand the formulas for Riemann sums

Left Riemann Sum: \(L_n = \Delta x \sum_{i=0}^{n-1} f(x_i)\) Right Riemann Sum: \(R_n = \Delta x \sum_{i=1}^{n} f(x_i)\) Midpoint Riemann Sum: \(M_n = \Delta x \sum_{i=0}^{n-1} f(x_i+\frac{\Delta x}{2})\)
02

Find the length of the subintervals

As the interval is divided into 4 equal subintervals, we can find the length by dividing the difference between the endpoints by the number of subintervals (n): \(\Delta x = \frac{6 - 2}{4} = \frac{4}{4} = 1\)
03

Find the Left Riemann Sum

Our Left Riemann Sum will be: \(L_4 = \Delta x \sum_{i=0}^{3} f(x_i) =1 \cdot [f(2) + f(3) + f(4) + f(5)] = [2^2 + 3^2 + 4^2 + 5^2]\)
04

Find the Right Riemann Sum

Our Right Riemann Sum will be: \(R_4 = \Delta x \sum_{i=1}^{4} f(x_i) =1 \cdot [f(3) + f(4) + f(5) + f(6)] = [3^2 + 4^2 + 5^2 + 6^2]\)
05

Find the Midpoint Riemann Sum

Our Midpoint Riemann Sum will be: \(M_4 = \Delta x \sum_{i=0}^{3} f(x_i+\frac{\Delta x}{2}) =1 \cdot [f(2.5) + f(3.5) + f(4.5) + f(5.5)] = [2.5^2 + 3.5^2 + 4.5^2 + 5.5^2]\) So, the left, right, and midpoint Riemann sums are: Left Riemann Sum: \(L_4 = [2^2 + 3^2 + 4^2 + 5^2]\) Right Riemann Sum: \(R_4 = [3^2 + 4^2 + 5^2 + 6^2]\) Midpoint Riemann Sum: \(M_4 = [2.5^2 + 3.5^2 + 4.5^2 + 5.5^2]\)

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