Textbook Question
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
โซโโด (4๐โ ๐ยฒ) d๐
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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 5.2.75a
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.
โซโโด 2โ๐ d๐
Verified step by step guidance1
Step 1: Understand the problem. A midpoint Riemann sum approximates the value of a definite integral by dividing the interval into n subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed.
Step 2: Define the interval and subintervals. The integral โซโโด 2โ๐ d๐ is over the interval [1, 4]. Divide this interval into n subintervals of equal width ฮ๐ = (4 - 1)/n = 3/n.
Step 3: Determine the midpoints of the subintervals. The midpoint of the i-th subinterval is given by ๐แตข = 1 + (i - 0.5)ฮ๐, where i ranges from 1 to n.
Step 4: Write the function value at the midpoint. The function being integrated is f(๐) = 2โ๐. At the midpoint ๐แตข, the function value is f(๐แตข) = 2โ(1 + (i - 0.5)ฮ๐).
Step 5: Express the midpoint Riemann sum in sigma notation. The sum is approximated as Sโ = ฮฃ (from i=1 to n) [f(๐แตข) * ฮ๐], which becomes Sโ = ฮฃ (from i=1 to n) [2โ(1 + (i - 0.5)(3/n)) * (3/n)].
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Riemann Sum
A Midpoint Riemann Sum is a method for approximating the value of a definite integral. It involves dividing the interval into 'n' subintervals, calculating the midpoint of each subinterval, and then evaluating the function at these midpoints. The sum of these values, multiplied by the width of the subintervals, provides an estimate of the area under the curve.
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Left, Right, & Midpoint Riemann Sums
Sigma Notation
Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (ฮฃ) to indicate summation, along with an index of summation that specifies the starting and ending values. In the context of Riemann sums, sigma notation allows us to express the sum of function values at midpoints over all subintervals in a clear and compact form.
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Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The definite integral provides a precise value that corresponds to the accumulation of quantities, such as area, over the interval, and is often evaluated using techniques like Riemann sums.
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Related Practice
Textbook Question
Area functions for linear functions Consider the following functions ฦ and real numbers a (see figure).
(a) Find and graph the area function A (๐) = โซโหฃ ฦ(t) dt .
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ฦ(t) = 4t + 2 , a = 0
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Textbook Question
Bounds on an integral Suppose ฦ is continuous on [a, b] with ฦ''(๐) > 0 on the interval. It can be shown that (bโa) ฦ [(a + b) /2] โค โซโแต ฦ(๐) d๐ โค (bโa) [ (ฦ(a) + ฦ(b)) /2]
(a) Assuming ฦ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b.
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Textbook Question
Matching functions with area functions Match the functions ฦ, whose graphs are given in aโ d, with the area functions A (๐) = โซโหฃ ฦ(t) dt, whose graphs are given in AโD.
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Textbook Question
Area functions The graph of ฦ is shown in the figure. Let A(x) = โซโโหฃ ฦ(t) dt and F(x) = โซโหฃ ฦ(t) dt be two area functions for ฦ. Evaluate the following area functions.
(a) A (โ2)
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Textbook Question
Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the ๐-direction and 2b in the y-direction is (๐ยฒ/aยฒ) + (yยฒ /bยฒ) = 1.
(a) Let dยฒ denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ โa, a] to show that the average value of dยฒ is (aยฒ + 2bยฒ) /3 .
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