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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.1.37d
Capitolo 5, Problema 5.1.37d

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


ƒ(𝓍) = 2x + 1 on [0,4] ; n = 4


d) Calculate the midpoint Riemann sum.

Guida verificata passo dopo passo
1
Step 1: Understand the problem. The midpoint Riemann sum is a method to approximate the area under a curve by dividing the interval into subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed. Here, the function is ƒ(𝓍) = 2x + 1, the interval is [0,4], and n = 4 (number of subintervals).
Step 2: Divide the interval [0,4] into n = 4 equal subintervals. The width of each subinterval, Δx, is calculated as Δx = (b - a) / n, where a = 0 and b = 4. Substitute the values to find Δx.
Step 3: Determine the midpoints of each subinterval. The midpoints are calculated as the average of the endpoints of each subinterval. For example, the first subinterval is [0,1], so its midpoint is (0 + 1) / 2 = 0.5. Repeat this for all subintervals.
Step 4: Evaluate the function ƒ(𝓍) = 2x + 1 at each midpoint. Substitute each midpoint value into the function to find the function values at those points.
Step 5: Multiply each function value by the width of the subinterval, Δx, and sum these products to calculate the midpoint Riemann sum. This sum represents the approximate area under the curve on the interval [0,4].

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Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over a specified interval. They involve dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals, and then summing the products of these values and the widths of the subintervals. The midpoint Riemann sum specifically uses the midpoint of each subinterval to evaluate the function.
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Introduction to Riemann Sums

Midpoint Rule

The midpoint rule is a specific type of Riemann sum that uses the midpoint of each subinterval to estimate the area under a curve. For a function f(x) over an interval [a, b] divided into n equal parts, the midpoint of each subinterval is calculated, and the function is evaluated at these midpoints. The sum of these values, multiplied by the width of the subintervals, provides an approximation of the integral.
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Definite Integral

A definite integral represents the signed area under a curve defined by a function f(x) between two points a and b on the x-axis. It is calculated as the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral provides a precise value for the area, which can be interpreted in various contexts, such as total distance, accumulated quantity, or net change.
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Domanda del libro di testo

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

f(x) = x + 1 on [0,4]; n = 4

(d) Calculate the left and right Riemann sums.                                                                                                                                                

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Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.


(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

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Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)

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{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..


∫₀² (𝓍²―2) d𝓍 ; n = 4

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{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.


∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

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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.

(d) F(4)

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