Textbook Question
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«ββΒΉ (πβ1) (πΒ²β2π)β· 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 5.2.59
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
Verified step by step guidance1
Identify the regions bounded by the graph of the function Ζ(π) and the π-axis between π = 0 and π = a. Note whether these regions lie above or below the π-axis, as this affects the sign of the integral.
Recall that the definite integral β«βα΅ Ζ(π) dπ represents the net area between the curve and the π-axis from 0 to a. Areas above the axis contribute positively, while areas below contribute negatively.
Determine the numerical values of the areas shown in the figure for each region between 0 and a. Assign positive values to areas above the axis and negative values to areas below the axis.
Sum all these signed areas to find the value of the definite integral β«βα΅ Ζ(π) dπ. This sum represents the net area under the curve from 0 to a.
Write the final expression for the integral as the sum of these areas, ensuring to include the correct signs based on the position relative to the π-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as Net Area
A definite integral from a to b of a function f(x) represents the net area between the graph of f and the x-axis over [a, b]. Areas above the x-axis contribute positively, while areas below contribute negatively, resulting in the integral's value.
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Definition of the Definite Integral
Interpreting Graphs for Integration
When evaluating integrals from graphs, it is essential to identify the regions bounded by the curve and the x-axis, noting their shapes and whether they lie above or below the axis. This helps in calculating areas accurately, often by summing or subtracting given region areas.
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Properties of Definite Integrals
Definite integrals have properties such as additivity over intervals and sign changes when limits are reversed. Understanding these properties allows breaking complex integrals into simpler parts and correctly combining areas to find the total integral value.
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Related Practice
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
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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Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dπ β«βΛ£ (β1 + tΒ²) dt (Hint: β«Λ£ββ (β1 + tΒ²) dt = β«β°ββ (β1 + tΒ²) dt + β«Λ£ββ (β1 + tΒ²) dt ) .
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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(p(π)) dπ