Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βα΅ Ζ(π) dπ
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 5.5.70
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· dπ
Verified step by step guidance1
Step 1: Recognize that the integral β«ββΒΉ (π - 1)(πΒ² - 2π)β· dπ involves a composite function (πΒ² - 2π) raised to a power. This suggests using a substitution method to simplify the integral.
Step 2: Let u = πΒ² - 2π. Compute the derivative of u with respect to π: du/dπ = 2π - 2. Rewrite this as du = (2π - 2)dπ.
Step 3: Factorize the original integral to match the substitution. Notice that (π - 1) can be factored out from the derivative du = (2π - 2)dπ, since (π - 1) is a common factor. Rewrite the integral in terms of u.
Step 4: Change the limits of integration. When π = -1, substitute into u = πΒ² - 2π to find the lower limit of u. Similarly, when π = 1, substitute into u = πΒ² - 2π to find the upper limit of u.
Step 5: Rewrite the integral entirely in terms of u and evaluate using standard integration techniques or a table of integrals. After integration, substitute back the original variable if necessary.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the accumulation of quantities, such as area, volume, or total change, between the limits of integration.
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Definition of the Definite Integral
Change of Variables
Change of variables, or substitution, is a technique used in integration to simplify the integrand by transforming it into a more manageable form. This involves substituting a new variable for an existing one, which can make the integral easier to evaluate. The process requires adjusting the limits of integration and the differential to maintain the integrity of the integral's value.
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Integration Techniques
Integration techniques encompass various methods used to evaluate integrals, including substitution, integration by parts, and using tables of integrals. These techniques are essential for solving complex integrals that cannot be evaluated using basic antiderivatives. Familiarity with these methods allows students to tackle a wide range of problems in calculus effectively.
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Related Practice
Textbook Question
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β« π cosΒ²πΒ² dπ
Textbook Question
Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.
β«α΅ββ Ζ(g(π)) dπ
Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« (sinβ΅ π + 3 sinΒ³ πβ sin π) cos π dπ
Textbook Question
The linear function Ζ(π) = 3 β π is decreasing on the interval [0, 3]. Is its area function for Ζ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain.
Textbook Question
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ and the π-axis. Evaluate the following integrals.
β«βαΆ |Ζ(π)| dπ
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