Textbook Question
Suppose Ζ is an odd function, β«ββ΄ Ζ(π) dπ = 3 , and β«ββΈ Ζ(π) dπ = 9 .
(a) Evaluate β«βββ΄ Ζ(π) dπ .
73
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 5.2.51a
Properties of integrals Use only the fact that β«ββ΄ 3π (4 βπ) dπ = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.
(a) β«ββ° 3π(4 β π) d(π)
Verified step by step guidance1
Step 1: Recognize the integral given in part (a) is the same as the integral provided in the problem, except the limits of integration are reversed. The integral provided is β«ββ΄ 3π(4 β π) dπ = 32.
Step 2: Recall the property of definite integrals: reversing the limits of integration changes the sign of the integral. Mathematically, β«βα΅ f(π) dπ = -β«α΅β f(π) dπ.
Step 3: Apply this property to the integral in part (a). Since the limits are reversed (from 4 to 0 instead of 0 to 4), the integral becomes -β«ββ΄ 3π(4 β π) dπ.
Step 4: Substitute the value of the original integral, which is given as 32. Therefore, the integral in part (a) becomes -32.
Step 5: Conclude that the integral β«ββ° 3π(4 β π) dπ evaluates to -32 based on the properties of integrals and the given information.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as β«βα΅ f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.
Recommended video:
05:43
Definition of the Definite Integral
Properties of Integrals
The properties of integrals include linearity, which allows for the integration of sums and scalar multiples, and the reversal of limits, which states that β«βα΅ f(x) dx = -β«α΅β f(x) dx. These properties enable the evaluation of integrals by transforming them into simpler forms or by changing the limits of integration.
Recommended video:
06:21Properties of Functions
Substitution in Integrals
Substitution is a technique used in integration to simplify the integrand by changing variables. This method involves selecting a new variable that simplifies the integral, allowing for easier computation. For example, if u = g(x), then dx can be expressed in terms of du, transforming the integral into a more manageable form.
Recommended video:
04:27Substitution With an Extra Variable
Related Practice
Textbook Question
Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earthβs surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)
(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints xβ = 34 , xβ = 40 , xβ = 46 , xβ = 52 , xβ = 58 , and xβ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.
64
views
Textbook Question
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«βΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(a) A(2)
66
views
Textbook Question
Area functions for the same linear function Let Ζ(t) = 2t β 2 and consider the two area functions A (π) = β«βΛ£ Ζ(t) dt and F(π) = β«βΛ£ Ζ(t) dt .
(a) Evaluate A (2) and A (3). Then use geometry to find an expression for A (π) , for π β₯ 1 .
70
views
Textbook Question
Substitutions Suppose Ζ is an even function with β«ββΈ Ζ(π) dπ = 9 . Evaluate each integral.
(a) β«ΒΉββ πΖ(πΒ²) dπ
44
views
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) If Ζ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of β«βα΅ Ζ(π) dπ, for any positive integer n.
66
views