Textbook Question
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
β«ββ΄ (4πβ πΒ²) 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 5.2.51a
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
Matching functions with area functions Match the functions Ζ, whose graphs are given in aβ d, with the area functions A (π) = β«βΛ£ Ζ(t) dt, whose graphs are given in AβD.
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 .
1
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
Textbook Question
Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the π-direction and 2b in the y-direction is (πΒ²/aΒ²) + (yΒ² /bΒ²) = 1.
(a) Let dΒ² denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ βa, a] to show that the average value of dΒ² is (aΒ² + 2bΒ²) /3 .
Textbook Question
Working with area functions Consider the function Ζ and the points a, b, and c.
(a) Find the area function A (π) = β«βΛ£ Ζ(t) dt using the Fundamental Theorem.
Ζ(π) = sin π ; a = 0 , b = Ο/2 , c = Ο