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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 5.2.75a
Capitolo 5, Problema 5.2.75a

{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

Guida verificata passo dopo passo
1
Step 1: Understand the problem. A midpoint Riemann sum approximates the value of a definite integral by dividing the interval into n subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed.
Step 2: Define the interval and subintervals. The integral ∫₁⁴ 2√𝓍 d𝓍 is over the interval [1, 4]. Divide this interval into n subintervals of equal width Δ𝓍 = (4 - 1)/n = 3/n.
Step 3: Determine the midpoints of the subintervals. The midpoint of the i-th subinterval is given by 𝓍ᵢ = 1 + (i - 0.5)Δ𝓍, where i ranges from 1 to n.
Step 4: Write the function value at the midpoint. The function being integrated is f(𝓍) = 2√𝓍. At the midpoint 𝓍ᵢ, the function value is f(𝓍ᵢ) = 2√(1 + (i - 0.5)Δ𝓍).
Step 5: Express the midpoint Riemann sum in sigma notation. The sum is approximated as Sₙ = Σ (from i=1 to n) [f(𝓍ᵢ) * Δ𝓍], which becomes Sₙ = Σ (from i=1 to n) [2√(1 + (i - 0.5)(3/n)) * (3/n)].

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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 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 allows us to express the sum of function values at midpoints over all subintervals in a clear and compact form.
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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 [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The definite integral provides a precise value that corresponds to the accumulation of quantities, such as area, over the interval, and is often evaluated using techniques like Riemann sums.
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Definition of the Definite Integral
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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.


∫₀⁴ (4𝓍― 𝓍²) d𝓍

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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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Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.



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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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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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