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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.1.25d
Chapter 5, Problem 5.1.25d

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
f(x) = x + 1 on [0,4]; n = 4
(d) Calculate the left and right Riemann sums.                                                                                                                                                

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Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function f(x) = x + 1 over the interval [0, 4] with n = 4 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Determine the width of each subinterval, Δx. The formula for Δx is Δx = (b - a) / n, where [a, b] is the interval and n is the number of subintervals. Here, a = 0, b = 4, and n = 4.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function. The left endpoints are x₀, x₁, x₂, ..., x₃. Calculate f(x) at each left endpoint and multiply by Δx. The sum is Σ f(xᵢ) * Δx for i = 0 to n-1.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function. The right endpoints are x₁, x₂, x₃, ..., x₄. Calculate f(x) at each right endpoint and multiply by Δx. The sum is Σ f(xᵢ) * Δx for i = 1 to n.
Step 5: Write out the expressions for both sums explicitly using the function f(x) = x + 1 and the calculated Δx. Substitute the values of the endpoints into the function and sum the results for both the left and right Riemann sums.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over a specified interval. They involve partitioning the interval into smaller subintervals, calculating the function's value at specific points (either left, right, or midpoints), and summing the products of these values and the widths of the subintervals. This technique provides a way to estimate the area under the curve of the function.
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Introduction to Riemann Sums

Left and Right Riemann Sums

Left and right Riemann sums are specific types of Riemann sums that use the leftmost and rightmost points of each subinterval, respectively, to evaluate the function. In a left Riemann sum, the function value at the left endpoint of each subinterval is used, while in a right Riemann sum, the function value at the right endpoint is used. These sums can yield different approximations of the integral, depending on the function's behavior over the interval.
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Left, Right, & Midpoint Riemann Sums

Partitioning the Interval

Partitioning the interval involves dividing the range of integration into 'n' equal subintervals, where 'n' is the number of partitions specified. For the function f(x) = x + 1 on the interval [0, 4] with n = 4, the interval is divided into four segments of equal width. This step is crucial for calculating Riemann sums, as it determines the points at which the function will be evaluated.
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Interval of Convergence
Related Practice
Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)  

Textbook Question

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.


(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


ƒ(𝓍) = 2x + 1 on [0,4] ; n = 4


d) Calculate the midpoint Riemann sum.

Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(e)     3                                                                                                                                                                               

       ∑  (2m + 2) / 3                                                                                                                                                                          

      m =1                         

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(4)