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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 5.2.23b
Capitolo 5, Problema 5.2.23b

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.


f(x) = sin 2x on [0,3π/4]


(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

Guida verificata passo dopo passo
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Step 1: Understand the problem. You are tasked with approximating the net area bounded by the graph of f(x) = sin(2x) and the x-axis on the interval [0, 3π/4] using three types of Riemann sums: left, right, and midpoint, with n = 4 subintervals.
Step 2: Divide the interval [0, 3π/4] into n = 4 equal subintervals. The width of each subinterval, Δx, is calculated as Δx = (3π/4 - 0)/4 = 3π/16.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate f(x). The left endpoints are x₀ = 0, x₁ = 3π/16, x₂ = 6π/16, and x₃ = 9π/16. Compute the sum: Left Sum ≈ Δx * [f(x₀) + f(x₁) + f(x₂) + f(x₃)].
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate f(x). The right endpoints are x₁ = 3π/16, x₂ = 6π/16, x₃ = 9π/16, and x₄ = 12π/16. Compute the sum: Right Sum ≈ Δx * [f(x₁) + f(x₂) + f(x₃) + f(x₄)].
Step 5: For the midpoint Riemann sum, use the midpoints of each subinterval to evaluate f(x). The midpoints are x₁ = (0 + 3π/16)/2, x₂ = (3π/16 + 6π/16)/2, x₃ = (6π/16 + 9π/16)/2, and x₄ = (9π/16 + 12π/16)/2. Compute the sum: Midpoint Sum ≈ Δx * [f(x₁) + f(x₂) + f(x₃) + f(x₄)].

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Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval. They involve dividing the interval into smaller subintervals, calculating the function's value at specific points within these subintervals (left endpoint, right endpoint, or midpoint), and then summing the areas of the rectangles formed. This technique helps estimate the net area under a curve, which is essential for understanding integration.
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Introduction to Riemann Sums

Definite Integral

The definite integral of a function over an interval provides the net area between the graph of the function and the x-axis. It accounts for both positive and negative areas, where areas above the x-axis contribute positively and those below contribute negatively. Understanding the concept of definite integrals is crucial for interpreting the results of Riemann sums and for calculating exact areas under curves.
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Definition of the Definite Integral

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, particularly when dealing with periodic functions. The function f(x) = sin(2x) oscillates between -1 and 1, and its behavior on the specified interval [0, 3π/4] affects the calculation of the net area. Recognizing the properties of these functions, including their periodicity and symmetry, is vital for accurately applying Riemann sums.
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Introduction to Trigonometric Functions
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{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.


(b) Calculate g'(𝓍)


g(𝓍) = ∫₀ˣ sin (πt² ) dt ( a Fresnel integral) 

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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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Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(b) ∫₃⁶ (―3g(𝓍)) d𝓍

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Working with area functions Consider the function ƒ and the points a, b, and c.

(b) Graph ƒ and A.

ƒ(𝓍) = 1/𝓍 ; a = 1 , b = 4 , c = 6

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The following functions are positive and negative on the given interval.

ƒ(𝓍) = xe⁻ˣ on [-1,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.

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{Use of Tech} Riemann sums for larger values of n Complete the following steps for the given function f and interval.


ƒ(𝓍) = x² ― 1 on [2,5] ; n = 75

(b) Based on the approximations found in part (a), estimate the area of the region bounded by the graph of f and the x-axis on the interval.

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