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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 5.1.53b
Chapter 5, Problem 5.1.53b

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


Ζ’(𝓍) = xΒ² ― 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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Step 1: Understand the problem. The goal is to estimate the area under the curve Ζ’(𝓍) = xΒ² ― 1 on the interval [2,5] using Riemann sums with n = 75 subintervals. This involves dividing the interval into 75 equal parts and summing the areas of rectangles formed under the curve.
Step 2: Calculate the width of each subinterval (Ξ”x). The formula for Ξ”x is Ξ”x = (b - a) / n, where [a, b] is the interval and n is the number of subintervals. Here, a = 2, b = 5, and n = 75.
Step 3: Determine the x-values for the subintervals. These are the points where the function will be evaluated. For a left Riemann sum, the x-values are xβ‚€, x₁, ..., xₙ₋₁, where xβ‚– = a + kΞ”x for k = 0, 1, ..., n-1. For a right Riemann sum, the x-values are x₁, xβ‚‚, ..., xβ‚™, where xβ‚– = a + kΞ”x for k = 1, 2, ..., n.
Step 4: Evaluate the function Ζ’(𝓍) = xΒ² ― 1 at each x-value. For each subinterval, calculate the height of the rectangle by substituting the x-value into the function Ζ’(𝓍).
Step 5: Compute the Riemann sum. Multiply the height of each rectangle (Ζ’(xβ‚–)) by the width of the subinterval (Ξ”x), and sum these products over all subintervals. This sum will approximate the area under the curve on the interval [2,5].

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Key Concepts

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

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval. By dividing the interval into 'n' subintervals and calculating the sum of the areas of rectangles formed by the function's values at specific points (left, right, or midpoints), we can estimate the total area under the curve. As 'n' increases, the approximation becomes more accurate, converging to the actual integral.
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Introduction to Riemann Sums

Definite Integral

The definite integral of a function over an interval represents the exact area under the curve of the function between two points. It is denoted as ∫[a,b] f(x) dx, where 'a' and 'b' are the limits of integration. The definite integral can be computed using various techniques, including Riemann sums, and is fundamental in calculus for determining total accumulation, such as area, volume, and displacement.
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Definition of the Definite Integral

Function Behavior

Understanding the behavior of the function f(x) = xΒ² - 1 is crucial for estimating the area under its curve. This quadratic function opens upwards and intersects the x-axis at x = -1 and x = 1, meaning it is positive on the interval [2, 5]. Analyzing the function's values within the specified interval helps in determining the area bounded by the graph and the x-axis, which is essential for accurate Riemann sum calculations.
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Graphs of Exponential Functions
Related Practice
Textbook Question

Suppose Ζ’ is an even function and βˆ«βΈβ‚‹β‚ˆ Ζ’(𝓍) d𝓍 = 18

(b) Evaluate βˆ«β‚‹β‚ˆβΈ 𝓍ƒ(𝓍) d𝓍 .

Textbook Question

{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) = βˆ«β‚€Λ£ sinΒ² t dt

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Textbook Question

{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) = βˆ«β‚€Λ£ sin (Ο€tΒ² ) dt ( a Fresnel integral) 

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Textbook Question

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = βˆ«β‚€ΒΉ (𝓍³ ― 2𝓍) d𝓍 = ―3/4

(b) βˆ«β‚β° (2𝓍―𝓍³) d𝓍

Textbook Question

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

Ζ’(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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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) A left Riemann sum always overestimates the area of a region bounded by a positive increasing function and the x-axis on an interval [a,b].

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