Textbook Question
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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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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(π) = cos π on [0. Ο/2]; n = 4
(d) Calculate the left and right Riemann sums.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function Ζ(π) = cos(π) 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, Ξπ, is calculated as Ξπ = (b - a) / n, where a = 0 and b = Ο/2. Using MathML: divided by 4.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function Ζ(π). The left endpoints are: πβ = 0, πβ = Ξπ, πβ = 2Ξπ, and πβ = 3Ξπ. Compute Ζ(π) = cos(π) at each of these points and multiply each value by Ξπ. 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(π). The right endpoints are: πβ = Ξπ, πβ = 2Ξπ, πβ = 3Ξπ, and πβ = 4Ξπ. Compute Ζ(π) = cos(π) at each of these points and multiply each value by Ξπ. 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 β«[a, b] f(x) dx, where [a, b] is the interval of integration.
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05:43Definition 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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5:53Graph of Sine and Cosine Function
Related Practice
Textbook Question
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(d) β«ββΆ (g(π) β f(π) dπ
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Textbook Question
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
Ζ(π) = 2x + 1 on [0,4] ; n = 4
d) Calculate the midpoint Riemann sum.
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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
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(d) β«ββΈ 3π(4 β π) d(π)
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Textbook Question
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} Ζ(π) = βx on [1,3] ; n = 4
(d) Calculate the midpoint Riemann sum.
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