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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 5.1.51b
Chapter 5, Problem 5.1.51b

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


ฦ’(๐“) = 3 โˆšx on [0,4] ; n = 40
(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. We are tasked with estimating the area under the curve of the function ฦ’(๐“) = 3โˆšx on the interval [0,4] using Riemann sums with n = 40 subintervals. This involves dividing the interval into smaller subintervals and summing the areas of rectangles under the curve.
Step 2: Divide the interval [0,4] into n = 40 subintervals. The width of each subinterval, ฮ”x, is calculated as ฮ”x = (b - a) / n, where a = 0 and b = 4. Substitute these values into the formula to find ฮ”x.
Step 3: Determine the x-values for the endpoints of each subinterval. These x-values are given by xโ‚€, xโ‚, xโ‚‚, ..., xโ‚™, where xโ‚€ = a and xโ‚™ = b. Use the formula xแตข = a + iฮ”x (for i = 0, 1, 2, ..., n) to calculate the x-values.
Step 4: Evaluate the function ฦ’(๐“) = 3โˆšx at the appropriate x-values. Depending on whether you are using left Riemann sums, right Riemann sums, or midpoint Riemann sums, evaluate ฦ’(๐“) at the left endpoints, right endpoints, or midpoints of each subinterval.
Step 5: Compute the Riemann sum. Multiply the function values by the width of the subintervals (ฮ”x) and sum them up. The formula for the Riemann sum is ฮฃ[ฦ’(xแตข)ฮ”x], where the summation runs over all subintervals. This sum provides an approximation of the area under the curve.

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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 area under a curve by dividing the interval into smaller subintervals. For each subinterval, a sample point is chosen, and the function value at that point is multiplied by the width of the subinterval. The sum of these products gives an estimate of the total area. As the number of subintervals (n) increases, the approximation becomes more accurate.
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Introduction to Riemann Sums

Definite Integrals

Definite integrals represent the exact area under a curve between two points on the x-axis. They are calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The definite integral of a function over an interval provides a precise value for the area, contrasting with Riemann sums, which provide an approximation that improves with larger n.
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Definition of the Definite Integral

Limit of Riemann Sums

The limit of Riemann sums as the number of subintervals approaches infinity leads to the exact value of the definite integral. This concept is crucial in calculus, as it formalizes the transition from discrete approximations to continuous areas. Understanding this limit helps in grasping how integration works and why it is a fundamental tool for calculating areas under curves.
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Introduction to Riemann Sums
Related Practice
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Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(b) Use geometry to find the displacement of the object between t = 0 and t = 2.

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{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.


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Working with area functions Consider the function ฦ’ and its graph.

(b) Estimate the points (if any) at which A has a local maximum or minimum.


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{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.

(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.


โˆซโ‚€ยน (๐“ยฒ + 1) d๐“

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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) If ฦ’ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of โˆซโ‚แต‡ ฦ’(๐“) d๐“, for any positive integer n. 

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Textbook Question

Working with area functions Consider the function ฦ’ and its graph.

(b) Estimate the points (if any) at which A has a local maximum or minimum.


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