Textbook Question
Sigma notation Evaluate the following expressions.
(c) 4
β ΞΊΒ²
ΞΊ=1
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 5.2.34c
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
β«β^Ο/2 cos π dπ ; n = 4
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with approximating the definite integral β«β^(Ο/2) cos(π) dπ using left and right Riemann sums 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 subintervals. The width of each subinterval, Ξπ, is calculated as Ξπ = (b - a) / n, where a = 0 and b = Ο/2. Substitute the values to find Ξπ.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function cos(π). The left endpoints are xβ = 0, xβ = Ξπ, xβ = 2Ξπ, and xβ = 3Ξπ. Compute the sum: Left Riemann Sum = Ξπ * [cos(xβ) + cos(xβ) + cos(xβ) + cos(xβ)].
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function cos(π). The right endpoints are xβ = Ξπ, xβ = 2Ξπ, xβ = 3Ξπ, and xβ = 4Ξπ. Compute the sum: Right Riemann Sum = Ξπ * [cos(xβ) + cos(xβ) + cos(xβ) + cos(xβ)].
Step 5: Compare the left and right Riemann sums to understand how the choice of endpoints affects the approximation. These sums provide an estimate of the integral β«β^(Ο/2) cos(π) dπ.
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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 value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles provides an estimate of the integral's value. Depending on whether the left or right endpoints of the subintervals are used, the sums can yield different approximations, which converge to the actual integral as the number of rectangles increases.
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Introduction to Riemann Sums
Definite Integral
A definite integral represents the signed area under a curve between two specified limits, often denoted as β«_a^b f(x) dx. It quantifies the accumulation of quantities, such as area, over an interval [a, b]. The Fundamental Theorem of Calculus connects differentiation and integration, stating that the definite integral can be evaluated using the antiderivative of the function.
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05:43Definition of the Definite Integral
Subintervals and n
In the context of Riemann sums, 'n' refers to the number of subintervals into which the interval [0, Ο/2] is divided. Each subinterval has a width of Ξx = (b - a)/n, which determines the height of the rectangles used in the approximation. A larger value of n results in narrower subintervals, leading to a more accurate approximation of the definite integral.
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06:11Introduction to Riemann Sums
Related Practice
Textbook Question
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(c) β«ββΆ (3Ζ(π) β g(π)) dπ
Textbook Question
Matching functions with area functions Match the functions Ζ, whose graphs are given in aβ d, with the area functions A (π) = β«βΛ£ Ζ(t) dt, whose graphs are given in AβD.
Textbook Question
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(c) Calculate the left and right Riemann sums for the given value of n.
β«βΒ² (πΒ²β2) dπ ; n = 4
Textbook Question
Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(c) Use geometry to find the displacement of the object between t = 2 and t = 5.
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Textbook Question
Approximating areas Estimate the area of the region bounded by the graph of Ζ(π) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.
(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.
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