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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 5.1.27d
Chapter 5, Problem 5.1.27d

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} Ζ’(𝓍) = cos 𝓍 on [0. Ο€/2]; n = 4
(d) Calculate the left and right Riemann sums.

Verified step by step guidance
1
Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function Ζ’(𝓍) = cos(𝓍) over the interval [0, Ο€/2] with n = 4 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [0, Ο€/2] into n = 4 equal subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 0 and b = Ο€/2. Using MathML: π2-0 divided by 4.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function Ζ’(𝓍). The left endpoints are: 𝓍₀ = 0, 𝓍₁ = Δ𝓍, 𝓍₂ = 2Δ𝓍, and 𝓍₃ = 3Δ𝓍. Compute Ζ’(𝓍) = cos(𝓍) at each of these points and multiply each value by Δ𝓍. Sum the results to get the left Riemann sum.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function Ζ’(𝓍). The right endpoints are: 𝓍₁ = Δ𝓍, 𝓍₂ = 2Δ𝓍, 𝓍₃ = 3Δ𝓍, and 𝓍₄ = 4Δ𝓍. Compute Ζ’(𝓍) = cos(𝓍) at each of these points and multiply each value by Δ𝓍. Sum the results to get the right Riemann sum.
Step 5: Summarize the process. The left Riemann sum uses the function values at the left endpoints, while the right Riemann sum uses the function values at the right endpoints. Both sums approximate the area under the curve, but they may differ slightly depending on the behavior of the function over the interval.

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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 dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals, and then summing the products of these values and the widths of the subintervals. The left Riemann sum uses the left endpoints of the subintervals, while the right Riemann sum uses the right endpoints.
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Introduction to Riemann Sums

Definite Integral

The definite integral of a function over an interval represents the net area under the curve of the function between two points. It is a fundamental concept in calculus that connects the concept of accumulation with the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is denoted as ∫[a, b] f(x) dx, where [a, b] is the interval of integration.
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Definition of the Definite Integral

Cosine Function

The cosine function, denoted as cos(x), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In calculus, it is important for understanding periodic functions and their integrals. The function cos(x) oscillates between -1 and 1, and its behavior over the interval [0, Ο€/2] is crucial for calculating the Riemann sums in the given problem.
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Graph of Sine and Cosine Function
Related Practice
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.


βˆ«β‚ƒβΆ (1―2𝓍) d𝓍 ; n = 6

Textbook Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t β‰₯ 5, find the function that gives the displacement between t = 0 and any time t β‰₯ 5.

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Textbook Question

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

{Use of Tech} Ζ’(𝓍) = e Λ£/β‚‚ on [1,4]; n = 6

(d) Calculate the left and right Riemann sums. 

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.                                                                          

                                                                                                                                                                                     (d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for Ζ’, then                                                                                                                                   

     B(𝓍) = 3𝓍² ― 𝓍 is also an area function for Ζ’.

Textbook Question

Area functions The graph of Ζ’ is shown in the figure. Let A(x) = βˆ«β‚€Λ£ Ζ’(t) dt and F(x) = βˆ«β‚‚Λ£ Ζ’(t) dt be two area functions for Ζ’. Evaluate the following area functions.

(d) F(8)

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Textbook Question

Sigma notation Evaluate the following expressions.

(d)     5                                                                                                                                                                              

       βˆ‘ (1 + nΒ²)                                                                                                                                                                          

       n=1