Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.R.9b

Chapter 5, Problem 5.R.9b

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .

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Step 1: Understand the problem. We are tasked with expressing the right Riemann sum for the integral ∫₁⁴ (3𝓍― 2) d𝓍 in terms of a positive integer n. A Riemann sum approximates the area under a curve by dividing the interval into subintervals and summing up the areas of rectangles.

Step 2: Define the interval and subintervals. The interval of integration is [1, 4]. Divide this interval into n subintervals of equal width Δ𝓍, where Δ𝓍 = (4 - 1)/n = 3/n.

Step 3: Determine the right endpoints of the subintervals. The right endpoint of the i-th subinterval is given by 𝓍ᵢ = 1 + iΔ𝓍, where i ranges from 1 to n.

Step 4: Write the function value at the right endpoint. The function to integrate is f(𝓍) = 3𝓍 - 2. At the right endpoint 𝓍ᵢ, the function value is f(𝓍ᵢ) = 3(1 + iΔ𝓍) - 2.

Step 5: Express the right Riemann sum in summation notation. The right Riemann sum is the sum of the areas of the rectangles, which is given by Sₙ = Σᵢ₌₁ⁿ f(𝓍ᵢ)Δ𝓍. Substituting f(𝓍ᵢ) and Δ𝓍, we get Sₙ = Σᵢ₌₁ⁿ [3(1 + i(3/n)) - 2](3/n).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into small rectangles. The sum of the areas of these rectangles provides an estimate of the integral. The choice of points within each subinterval (left, right, or midpoint) affects the accuracy of the approximation.

Definite Integral

A definite integral represents the signed area under a curve between two specified limits, often denoted as ∫ₐᵇ f(x) dx. It quantifies the accumulation of quantities, such as area, over an interval. The Fundamental Theorem of Calculus links the concept of differentiation with integration, allowing for the evaluation of definite integrals using antiderivatives.

Summation Notation

Summation notation, represented by the sigma symbol (Σ), is a concise way to express the sum of a sequence of terms. In the context of Riemann sums, it is used to represent the total area of rectangles formed by evaluating the function at specific points within subintervals. This notation simplifies the expression of sums, especially as the number of subdivisions increases.

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Sigma Notation

Related Practice

Textbook Question

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 16―𝓍² on [―4, 4]

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Textbook Question

Velocity to displacement An object travels on the 𝓍-axis with a velocity given by v(t) = 2t + 5, for 0 ≤ t ≤ 4.

(a) How far does the object travel, for 0 ≤ t ≤ 4 ?

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Textbook Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

―∫₄¹ 2ƒ(𝓍) d𝓍

Textbook Question

Evaluating integrals Evaluate the following integrals.

∫π/₆^π/³ (sec² t + csc² t) dt

Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t if ―2 ≤ t < 0

t²/2 if 0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .

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Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .

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Key Concepts

Riemann Sums

Definite Integral

Summation Notation

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .