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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.1.69c
Chapter 5, Problem 5.1.69c

Approximating areas Estimate the area of the region bounded by the graph of Ζ’(𝓍) = xΒ² + 2 and the x-axis on [0, 2] in the following ways.
(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

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Step 1: Understand the problem. We are tasked with estimating the area under the curve Ζ’(𝓍) = xΒ² + 2 on the interval [0, 2] using a right Riemann sum with n = 4 subintervals. A Riemann sum approximates the area under a curve by summing the areas of rectangles formed over subintervals.
Step 2: Divide the interval [0, 2] into n = 4 subintervals. The width of each subinterval, Ξ”x, is calculated as Ξ”x = (2 - 0) / 4 = 0.5. The subintervals are [0, 0.5], [0.5, 1], [1, 1.5], and [1.5, 2].
Step 3: For a right Riemann sum, the height of each rectangle is determined by the function value at the right endpoint of each subinterval. The right endpoints of the subintervals are x = 0.5, x = 1, x = 1.5, and x = 2. Evaluate Ζ’(𝓍) = xΒ² + 2 at these points: Ζ’(0.5), Ζ’(1), Ζ’(1.5), and Ζ’(2).
Step 4: Compute the area of each rectangle. The area of a rectangle is given by height Γ— width. For each subinterval, the width is Ξ”x = 0.5, and the height is the function value at the right endpoint. The areas are: A₁ = Ζ’(0.5) Γ— 0.5, Aβ‚‚ = Ζ’(1) Γ— 0.5, A₃ = Ζ’(1.5) Γ— 0.5, and Aβ‚„ = Ζ’(2) Γ— 0.5.
Step 5: Add the areas of all rectangles to approximate the total area under the curve. The right Riemann sum is given by: Rβ‚„ = A₁ + Aβ‚‚ + A₃ + Aβ‚„. This sum provides the estimated area of the region bounded by the graph of Ζ’(𝓍) = xΒ² + 2 and the x-axis on [0, 2].

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve on a given interval by dividing the area into smaller subintervals. Each subinterval's area is estimated using the value of the function at specific points, such as the right endpoint, left endpoint, or midpoint. The sum of these areas provides an approximation of the integral of the function over the interval.
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Subintervals

Subintervals are smaller segments into which a larger interval is divided to facilitate the approximation of areas or integrals. In this case, the interval [0, 2] is divided into four equal parts, each of width Ξ”x = (2-0)/4 = 0.5. This division allows for a more manageable calculation of the area under the curve by evaluating the function at specific points within each subinterval.
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Introduction to Riemann Sums

Geometric Interpretation

The geometric interpretation of a Riemann sum involves visualizing the area under the curve as a series of rectangles. For a right Riemann sum, the height of each rectangle is determined by the function's value at the right endpoint of each subinterval. This visual representation helps in understanding how the approximation converges to the actual area as the number of subintervals increases.
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Related Practice
Textbook Question

Properties of integrals Suppose βˆ«β‚€Β³Ζ’(𝓍) d𝓍 = 2 , βˆ«β‚ƒβΆΖ’(𝓍) d𝓍 = ―5 , and βˆ«β‚ƒβΆg(𝓍) d𝓍 = 1. Evaluate the following integrals.

(c) βˆ«β‚ƒβΆ (3Ζ’(𝓍) ― g(𝓍)) d𝓍

Textbook Question

Matching functions with area functions Match the functions Ζ’, whose graphs are given in a― d, with the area functions A (𝓍) = βˆ«β‚€Λ£ Ζ’(t) dt, whose graphs are given in A–D.



Textbook Question

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

(c) Calculate the left and right Riemann sums for the given value of n.


βˆ«β‚€Β² (𝓍²―2) d𝓍 ; n = 4

Textbook Question

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

(c) Use geometry to find the displacement of the object between t = 2 and t = 5.

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Textbook Question

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

(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .

Ζ’(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Textbook Question

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

(c) Calculate the left and right Riemann sums for the given value of n.


βˆ«β‚€^Ο€/2 cos 𝓍 d𝓍 ; n = 4

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