Textbook Question
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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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.1.37d
Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.
Ζ(π) = 2x + 1 on [0,4] ; n = 4
d) Calculate the midpoint Riemann sum.
Verified step by step guidance1
Step 1: Understand the problem. The midpoint Riemann sum is a method to approximate the area under a curve by dividing the interval into subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed. Here, the function is Ζ(π) = 2x + 1, the interval is [0,4], and n = 4 (number of subintervals).
Step 2: Divide the interval [0,4] into n = 4 equal subintervals. The width of each subinterval, Ξx, is calculated as Ξx = (b - a) / n, where a = 0 and b = 4. Substitute the values to find Ξx.
Step 3: Determine the midpoints of each subinterval. The midpoints are calculated as the average of the endpoints of each subinterval. For example, the first subinterval is [0,1], so its midpoint is (0 + 1) / 2 = 0.5. Repeat this for all subintervals.
Step 4: Evaluate the function Ζ(π) = 2x + 1 at each midpoint. Substitute each midpoint value into the function to find the function values at those points.
Step 5: Multiply each function value by the width of the subinterval, Ξx, and sum these products to calculate the midpoint Riemann sum. This sum represents the approximate area under the curve on the interval [0,4].
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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 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 midpoint Riemann sum specifically uses the midpoint of each subinterval to evaluate the function.
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Introduction to Riemann Sums
Midpoint Rule
The midpoint rule is a specific type of Riemann sum that uses the midpoint of each subinterval to estimate the area under a curve. For a function f(x) over an interval [a, b] divided into n equal parts, the midpoint of each subinterval is calculated, and the function is evaluated at these midpoints. The sum of these values, multiplied by the width of the subintervals, provides an approximation of the integral.
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Guided course
5:50Power Rules
Definite Integral
A definite integral represents the signed area under a curve defined by a function f(x) between two points a and b on the x-axis. It is calculated as the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral provides a precise value for the area, which can be interpreted in various contexts, such as total distance, accumulated quantity, or net change.
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Related Practice
Textbook Question
Properties of integrals Consider two functions Ζ and g on [1,6] such that β«ββΆΖ(π) dπ = 10 and β«ββΆg(π) dπ = 5, β«ββΆΖ(π) dπ = 5 , and β«ββ΄g(π) dπ = 2. Evaluate the following integrals.
(d) β«ββΆ (g(π) β f(π) dπ
Textbook Question
Matching functions with area functions Match the functions Ζ, whose graphs are given in aβ d, with the area functions A (π) = β«βΛ£ Ζ(t) dt, whose graphs are given in AβD.
Textbook Question
{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.
(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..
β«βΒ² (πΒ²β2) dπ ; n = 4
Textbook Question
Area functions The graph of Ζ is shown in the figure. Let A(x) = β«ββΛ£ Ζ(t) dt and F(x) = β«βΛ£ Ζ(t) dt be two area functions for Ζ. Evaluate the following area functions.
(d) F(4)
Textbook Question
Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(d) 1 + 1/2 + 1/3 + 1/4