Textbook Question
Use Table 5.6 to evaluate the following definite integrals.
(d) β«β^Ο/ΒΉβΆ sec Β² 4π dπ
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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 5.2.33d
{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.
β«ββ· 1/π dπ ; n = 6
Verified step by step guidance1
Step 1: Understand the problem. The integral β«ββ· (1/π) dπ represents the area under the curve of the function f(π) = 1/π from x = 1 to x = 7. We are tasked with determining which Riemann sum (left or right) underestimates or overestimates the value of this definite integral, given n = 6 subintervals.
Step 2: Divide the interval [1, 7] into n = 6 subintervals. The width of each subinterval, Ξπ, is calculated as Ξπ = (7 - 1)/6 = 6/6 = 1. This means the subintervals are [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], and [6, 7].
Step 3: Recall the definitions of the left and right Riemann sums. The left Riemann sum uses the function values at the left endpoints of each subinterval, while the right Riemann sum uses the function values at the right endpoints of each subinterval. For f(π) = 1/π, the left endpoints are {1, 2, 3, 4, 5, 6}, and the right endpoints are {2, 3, 4, 5, 6, 7}.
Step 4: Analyze the behavior of the function f(π) = 1/π. Since f(π) is a decreasing function on the interval [1, 7], the left Riemann sum will underestimate the area (because it uses smaller function values at the left endpoints), and the right Riemann sum will overestimate the area (because it uses larger function values at the right endpoints).
Step 5: Conclude that for the given integral β«ββ· (1/π) dπ and n = 6, the left Riemann sum underestimates the value of the definite integral, while the right Riemann sum overestimates the value of the definite integral.
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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 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 Riemann sum can either overestimate or underestimate the actual area, which is crucial for understanding the behavior of the integral.
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Introduction to Riemann Sums
Definite Integrals
A definite integral represents the net area under a curve between two specified limits, providing a precise value for the accumulation of quantities. It is denoted as β«βα΅ f(x) dx, where 'a' and 'b' are the limits of integration. Understanding how to evaluate definite integrals is essential for applying Riemann sums effectively, as it allows for comparison between the approximated and actual values.
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05:43Definition of the Definite Integral
Underestimation and Overestimation
In the context of Riemann sums, underestimation occurs when the sum of the areas of rectangles is less than the actual area under the curve, while overestimation occurs when it is greater. For a decreasing function, the left Riemann sum underestimates and the right Riemann sum overestimates the integral. Conversely, for an increasing function, the left sum overestimates and the right sum underestimates, making it important to analyze the function's behavior to determine which sum to use.
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10:22Left, Right, & Midpoint Riemann Sums Example 1
Related Practice
Textbook Question
Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
Ζ(π) = xΒ² β 1 on [2,4]; n = 4
(d) Calculate the left and right Riemann sums.
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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
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Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The functions p(π) = sin 3π and q(π) = 4 sin 3π are antiderivatives of the same function.
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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.
(a) β«βΒ³ 5Ζ(π) dπ
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Textbook Question
Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(c) Find the mass of the entire rod (0 β€ x β€ 10) .
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