Textbook Question
Working with area functions Consider the function ฦ and its graph.
(c) Sketch a graph of A, for 0 โค ๐ โค 10 , without a scale on the y-axis.
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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 5.2.33c
{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.
โซโโท 1/๐ d๐ ; n = 6
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with approximating the definite integral โซโโท (1/๐) d๐ using left and right Riemann sums with n = 6 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [1, 7] into n = 6 subintervals. The width of each subinterval, ฮ๐, is calculated as ฮ๐ = (7 - 1) / 6 = 6 / 6 = 1.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function f(๐) = 1/๐. The left endpoints are ๐โ = 1, ๐โ = 2, ๐โ = 3, ๐โ = 4, ๐โ = 5, and ๐โ
= 6. Compute the sum: Left Riemann Sum = ฮ๐ * [f(๐โ) + f(๐โ) + f(๐โ) + f(๐โ) + f(๐โ) + f(๐โ
)].
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function f(๐) = 1/๐. The right endpoints are ๐โ = 2, ๐โ = 3, ๐โ = 4, ๐โ = 5, ๐โ
= 6, and ๐โ = 7. Compute the sum: Right Riemann Sum = ฮ๐ * [f(๐โ) + f(๐โ) + f(๐โ) + f(๐โ) + f(๐โ
) + f(๐โ)].
Step 5: After setting up the sums, substitute the values of f(๐) = 1/๐ for each endpoint into the formulas for the left and right Riemann sums. Simplify the expressions to approximate the 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 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 โซโแต 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 of integration is divided. Each subinterval has a width of ฮx, calculated as (b - a)/n. The choice of n affects the accuracy of the approximation; a larger n results in narrower rectangles and a more precise estimate of the integral, while a smaller n may lead to greater error.
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Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .
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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.
โซโโถ (1โ2๐) d๐ ; n = 6
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Textbook Question
Zero net area Consider the function ฦ(๐) = ๐ยฒ โ 4๐ .
c) In general, for the function ฦ(๐) = ๐ยฒ โ a๐, where a > 0, for what value of b > 0 (as a function of a) is โซโแต ฦ(๐) d๐ = 0 ?
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Textbook Question
Use Table 5.6 to evaluate the following definite integrals.
(c) โซโโโ^โถ d๐/(๐ยฒ โ9)
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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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