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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.2.77a
Capitolo 5, Problema 5.2.77a

{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.


∫₀⁴ (4𝓍― 𝓍²) d𝓍

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Step 1: Understand the problem. A midpoint Riemann sum is a method to approximate the value of a definite integral by dividing the interval into subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed. The goal is to express this sum in sigma notation for an arbitrary number of subintervals, n.
Step 2: Define the interval and subintervals. The integral ∫₀⁴ (4𝓍 - 𝓍²) d𝓍 is over the interval [0, 4]. Divide this interval into n subintervals of equal width, Δ𝓍 = (4 - 0)/n = 4/n.
Step 3: Determine the midpoints of the subintervals. The midpoints of the subintervals are given by 𝓍ᵢ = a + (i - 0.5)Δ𝓍, where a = 0 is the starting point of the interval, i is the index of the subinterval (ranging from 1 to n), and Δ𝓍 = 4/n.
Step 4: Write the function value at the midpoints. For each midpoint 𝓍ᵢ, evaluate the function f(𝓍) = 4𝓍 - 𝓍². Substitute 𝓍ᵢ into the function to get f(𝓍ᵢ) = 4(0 + (i - 0.5)(4/n)) - (0 + (i - 0.5)(4/n))².
Step 5: Write the midpoint Riemann sum in sigma notation. The sum is given by Sₙ = Σᵢ₌₁ⁿ f(𝓍ᵢ)Δ𝓍, where Δ𝓍 = 4/n and f(𝓍ᵢ) is the function value at the midpoint. Substitute Δ𝓍 and f(𝓍ᵢ) into the formula to express the sum in terms of n and i.

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Midpoint Riemann Sum

A Midpoint Riemann Sum is a method for approximating the value of a definite integral. It involves dividing the interval into 'n' subintervals, calculating the midpoint of each subinterval, and then evaluating the function at these midpoints. The sum of these function values, multiplied by the width of the subintervals, provides an estimate of the area under the curve.
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Left, Right, & Midpoint Riemann Sums

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, along with an index of summation that specifies the starting and ending values. In the context of Riemann sums, sigma notation is used to express the sum of function values at midpoints across all subintervals.
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Sigma Notation

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫ from 'a' to 'b' of f(x) dx, where 'a' and 'b' are the limits of integration. The definite integral can be approximated using Riemann sums, which provide a numerical method to estimate the area when the exact integral is difficult to compute.
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Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).                                                                                           

                                                                                                                                                                                     

 (a) Find and graph the area function A (𝓍) = ∫ₐˣ ƒ(t) dt .                                                                                                                               

                                                                                                                                                                               

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 ƒ(t) = 4t + 2 , a = 0

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Bounds on an integral Suppose ƒ is continuous on [a, b] with ƒ''(𝓍) > 0 on the interval. It can be shown that (b―a) ƒ [(a + b) /2] ≤ ∫ₐᵇ ƒ(𝓍) d𝓍 ≤ (b―a) [ (ƒ(a) + ƒ(b)) /2]                                                         

                                                                                                                                                                               

(a) Assuming ƒ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. 

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Working with area functions Consider the function ƒ and its graph.

(a) Estimate the zeros of the area function A(𝓍) = ∫₀ˣ ƒ(t) dt , for 0 ≤ 𝓍 ≤ 10 .


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Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(a) A (―2)

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{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.


∫₁⁴ 2√𝓍 d𝓍

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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 𝓍-direction and 2b in the y-direction is (𝓍²/a²) + (y² /b²) = 1.

(a) Let d² denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ ―a, a] to show that the average value of d² is (a² + 2b²) /3 .

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