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.
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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 5.2.31d
{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«ββΆ (1β2π) dπ ; n = 6
Verified step by step guidance1
Step 1: Understand the problem. The integral β«ββΆ (1 - 2π) dπ represents the area under the curve of the function f(π) = 1 - 2π from x = 3 to x = 6. We are tasked with determining which Riemann sum (left or right) underestimates or overestimates the value of the definite integral, given n = 6 subintervals.
Step 2: Divide the interval [3, 6] into n = 6 subintervals. The width of each subinterval, Ξπ, is calculated as Ξπ = (b - a) / n, where a = 3 and b = 6. Substitute the values to find Ξπ.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to approximate the integral. The formula for the left Riemann sum is: Sβ = Ξ£ [f(πα΅’) * Ξπ], where πα΅’ represents the left endpoint of each subinterval. Compute the left endpoints and set up the summation.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to approximate the integral. The formula for the right Riemann sum is: Sα΅£ = Ξ£ [f(πα΅’) * Ξπ], where πα΅’ represents the right endpoint of each subinterval. Compute the right endpoints and set up the summation.
Step 5: Analyze the behavior of the function f(π) = 1 - 2π. Since f(π) is a decreasing linear function, the left Riemann sum will underestimate the integral (as it uses lower values of f(π)), while the right Riemann sum will overestimate the integral (as it uses higher values of f(π)).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The notation β«βα΅ f(x) dx indicates the integral of f(x) from a to b, providing a numerical value that reflects the net area between the curve and the x-axis.
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Definition of the Definite Integral
Riemann Sum
A Riemann sum is a method for approximating the value of a definite integral by dividing the area under the curve into smaller rectangles. The sum can be calculated using left endpoints, right endpoints, or midpoints of the subintervals. The choice of endpoints affects whether the sum underestimates or overestimates the integral, depending on the function's behavior over the interval.
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06:11Introduction to Riemann Sums
Underestimation and Overestimation
In the context of Riemann sums, underestimation occurs when the chosen rectangles fall below the curve, while overestimation happens when they extend above it. For a decreasing function, the left Riemann sum will typically overestimate the integral, and the right sum will underestimate it. Conversely, for an increasing function, the left sum will underestimate, and the right sum will overestimate the integral.
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10:22Left, Right, & Midpoint Riemann Sums Example 1
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(d) If β«βα΅ Ζ(π) dπ = β«βα΅ Ζ(π) dπ, then Ζ is a constant 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.
(d) β«βΒ³ (Ζ(π) + 2g(π)) dπ
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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.
(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«βΒ² (πΒ²β2) dπ ; n = 4
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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.
(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«β^Ο/2 cos π dπ ; n = 4
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Textbook Question
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«ββΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(d) F(4)
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