Textbook Question
Area by geometry Use geometry to evaluate the following definite integrals, where the graph of Ζ is given in the figure.
(b) β«ββ΄ Ζ(π) dπ
1
views
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.R.18
Properties of integrals Suppose β«ββ΄ Ζ(π) dπ = 6 , β«ββ΄ g(π) dπ = 4 and β«ββ΄ Ζ(π) dπ = 2 . Evaluate the following integrals or state that there is not enough information.
ββ«βΒΉ 2Ζ(π) dπ
Verified step by step guidance1
Step 1: Recognize that the integral β«βΒΉ 2Ζ(π) dπ involves reversing the limits of integration. When the limits are reversed, the integral changes sign. Thus, β«βΒΉ 2Ζ(π) dπ = -β«ββ΄ 2Ζ(π) dπ.
Step 2: Use the property of integrals that allows constants to be factored out. Specifically, β«ββ΄ 2Ζ(π) dπ = 2β«ββ΄ Ζ(π) dπ.
Step 3: Substitute the given value of β«ββ΄ Ζ(π) dπ = 6 into the equation from Step 2. This gives β«ββ΄ 2Ζ(π) dπ = 2 Γ 6.
Step 4: Combine the results from Step 1 and Step 3 to express the integral as -β«ββ΄ 2Ζ(π) dπ = -(2 Γ 6).
Step 5: Conclude that the integral β«βΒΉ 2Ζ(π) dπ can be evaluated using the steps above, but the final numerical result is not calculated here as per the instructions.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Definite Integrals
Definite integrals have several key properties, including linearity and the ability to reverse limits. The linearity property states that β«[a,b] (c * f(x)) dx = c * β«[a,b] f(x) dx for any constant c. Additionally, reversing the limits of integration changes the sign: β«[b,a] f(x) dx = -β«[a,b] f(x) dx. Understanding these properties is essential for evaluating integrals efficiently.
Recommended video:
Guided course
05:43
Definition of the Definite Integral
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then β«[a,b] f(x) dx = F(b) - F(a). This theorem allows us to evaluate definite integrals by finding the antiderivative, which is crucial for solving integral problems and understanding the relationship between the two operations.
Recommended video:
Guided course
06:11Fundamental Theorem of Calculus Part 1
Substitution in Integrals
Substitution is a technique used in integration to simplify the process of evaluating integrals. It involves changing the variable of integration to make the integral easier to solve. For example, if we let u = g(x), then the integral β« f(g(x)) g'(x) dx can be transformed into β« f(u) du, which may be simpler to evaluate. This concept is particularly useful when dealing with composite functions.
Recommended video:
04:27Substitution With an Extra Variable
Related Practice
Textbook Question
Find the intervals on which Ζ(π) = β«βΒΉ (tβ3) (tβ6)ΒΉΒΉ dt is increasing and the intervals on which it is decreasing.
1
views
Textbook Question
Evaluate the following derivatives.
d/dπ β«βα΅Λ£ cos tΒ² dt
1
views
Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(e) β«ββΒ² 3πΖ(π)dπ
Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(a) β«βββ΄ Ζ(π) dπ
Textbook Question
Symmetry properties Suppose β«ββ΄ Ζ(π) dπ = 10 and β«ββ΄ g(π) dπ = 20. Furthermore, suppose Ζ is an even function and g is an odd function. Evaluate the following integrals.
(c) β«βββ΄ (4Ζ(π) β 3g(π))dπ