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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 5 - Integration문제 5.1.27d
5장, 문제 5.1.27d

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4
(d) Calculate the left and right Riemann sums.

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Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function ƒ(𝓍) = cos(𝓍) over the interval [0, π/2] with n = 4 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.
Step 2: Divide the interval [0, π/2] into n = 4 equal subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 0 and b = π/2. Using MathML: π2-0 divided by 4.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function ƒ(𝓍). The left endpoints are: 𝓍₀ = 0, 𝓍₁ = Δ𝓍, 𝓍₂ = 2Δ𝓍, and 𝓍₃ = 3Δ𝓍. Compute ƒ(𝓍) = cos(𝓍) at each of these points and multiply each value by Δ𝓍. Sum the results to get the left Riemann sum.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function ƒ(𝓍). The right endpoints are: 𝓍₁ = Δ𝓍, 𝓍₂ = 2Δ𝓍, 𝓍₃ = 3Δ𝓍, and 𝓍₄ = 4Δ𝓍. Compute ƒ(𝓍) = cos(𝓍) at each of these points and multiply each value by Δ𝓍. Sum the results to get the right Riemann sum.
Step 5: Summarize the process. The left Riemann sum uses the function values at the left endpoints, while the right Riemann sum uses the function values at the right endpoints. Both sums approximate the area under the curve, but they may differ slightly depending on the behavior of the function over the interval.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 left Riemann sum uses the left endpoints of the subintervals, while the right Riemann sum uses the right endpoints.
추천 영상:
가이드 코스
06:11
Introduction to Riemann Sums

Definite Integral

The definite integral of a function over an interval represents the net area under the curve of the function between two points. It is a fundamental concept in calculus that connects the concept of accumulation with the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is denoted as ∫[a, b] f(x) dx, where [a, b] is the interval of integration.
추천 영상:
가이드 코스
05:43
Definition of the Definite Integral

Cosine Function

The cosine function, denoted as cos(x), is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. In calculus, it is important for understanding periodic functions and their integrals. The function cos(x) oscillates between -1 and 1, and its behavior over the interval [0, π/2] is crucial for calculating the Riemann sums in the given problem.
추천 영상:
가이드 코스
5:53
Graph of Sine and Cosine Function
관련 실천
교과서 질문

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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교과서 질문

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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교과서 질문

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.

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교과서 질문

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

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교과서 질문

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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교과서 질문

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


{Use of Tech} ƒ(𝓍) = √x on [1,3] ; n = 4


(d) Calculate the midpoint Riemann sum.

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