Textbook Question
Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = 𝓍³ on [―1, 1]
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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 5.1.65
Identifying Riemann sums Fill in the blanks with an interval and a value of n.
4
∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with
k = 1
n = ________ .
Verified step by step guidance1
Step 1: Understand the structure of the Riemann sum. A Riemann sum approximates the area under a curve by summing up the areas of rectangles. The given sum is a right Riemann sum, which means the function values are evaluated at the right endpoints of the subintervals.
Step 2: Analyze the given sum. The expression ƒ(1 + k) • 1 suggests that the interval starts at 1, and the width of each subinterval (Δx) is 1. The term 'k' represents the index of the subintervals.
Step 3: Determine the interval. Since the sum starts at k = 1 and increments by 1 for each term, the interval can be determined by the range of values for x. The interval is [1, 1 + n], where n is the number of subintervals.
Step 4: Identify the value of n. The summation symbol (∑) indicates that there are 4 terms in the sum, which means n = 4. This corresponds to the number of subintervals used in the approximation.
Step 5: Fill in the blanks. Based on the analysis, the interval is [1, 5] (since n = 4 and the interval ends at 1 + n), and the value of n is 4.
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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 an interval. They involve dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals, and summing the products of these values and the widths of the subintervals. The choice of points (left, right, or midpoint) affects the approximation's accuracy.
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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 calculated as the limit of Riemann sums as the number of subintervals approaches infinity. This concept is fundamental in calculus, linking the idea of accumulation with the geometric interpretation of area.
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05:43Definition of the Definite Integral
Interval Notation
Interval notation is a mathematical notation used to represent a range of values. It specifies the lower and upper bounds of the interval, using brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints. Understanding interval notation is crucial for correctly identifying the limits of integration when working with Riemann sums and definite integrals.
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Related Practice
Textbook Question
Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.
The region between the graph of y = 1 - |x| and the x-axis, for -2 ≤ x ≤ 2
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Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.
∫ₐ⁰ ƒ(𝓍) d𝓍
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Textbook Question
Variations on the substitution method Evaluate the following integrals.
∫ (eˣ ― e⁻ˣ)/ (eˣ + e⁻ˣ) d𝓍
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Textbook Question
Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).
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Textbook Question
Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = cos 𝓍 on [―π/2 , π/2]
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