Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure.
β«βΒΉ (πΒ² β 2π + 3) dπ
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 5.2.35
Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function Ζ on [a,b]. Identify Ζ and express the limit as a definite integral.
n
lim β (πβ*Β² + 1) βπβ on [0,2]
β β 0 k=1
Verified step by step guidance1
Step 1: Recognize the structure of the given limit. The expression lim β (πβΒ² + 1) βπβ as β β 0 represents a Riemann sum. A Riemann sum approximates the area under a curve by summing up the areas of rectangles, where the height of each rectangle is determined by the function value at a specific point within the interval.
Step 2: Identify the function Ζ(π) from the given Riemann sum. The term (πβΒ² + 1) corresponds to the function Ζ(π) = πΒ² + 1. This is the function being integrated over the interval [0, 2].
Step 3: Determine the interval of integration. The problem specifies the interval [0, 2], which means the definite integral will be evaluated from π = 0 to π = 2.
Step 4: Express the limit as a definite integral. The limit of the Riemann sum as β β 0 is equivalent to the definite integral of the function Ζ(π) = πΒ² + 1 over the interval [0, 2]. Using integral notation, this is written as:
Step 5: Conclude the process. The definite integral represents the exact area under the curve Ζ(π) = πΒ² + 1 from π = 0 to π = 2.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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 by dividing the interval into smaller subintervals. For each subinterval, a sample point is chosen, and the function's value at that point is multiplied by the width of the subinterval. As the number of subintervals increases and their width decreases, the Riemann sum approaches the exact value of the definite integral.
Recommended video:
06:11
Introduction to Riemann Sums
Definite Integrals
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as β«[a,b] f(x) dx and can be interpreted as the limit of Riemann sums as the number of subintervals approaches infinity. Definite integrals have numerous applications in calculating areas, volumes, and solving problems in physics and engineering.
Recommended video:
05:43Definition of the Definite Integral
Limits
In calculus, a limit describes the behavior of a function as its input approaches a certain value. When evaluating Riemann sums, the limit is taken as the width of the subintervals approaches zero, which allows for the transition from a sum of areas of rectangles to the exact area under the curve. Understanding limits is crucial for grasping the foundational concepts of continuity, derivatives, and integrals.
Recommended video:
05:50One-Sided Limits
Related Practice
Textbook Question
Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).
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.
Ζ(π) = cos π on [βΟ/2 , Ο/2]
1
views
Textbook Question
If Ζ is an odd function, why is β«α΅ββ Ζ(π) dπ = 0?
Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dπ β«βΛ£ (tΒ² + t + 1) dt
1
views
Textbook Question
Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.
Ζ(π) = 1 β |π| on [β1, 1]
1
views