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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 5.2.75b
Chapter 5, Problem 5.2.75b

{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.


βˆ«β‚β΄ 2βˆšπ“ d𝓍

Verified step by step guidance
1
Step 1: Understand the problem. The goal is to estimate the value of the definite integral βˆ«β‚β΄ 2βˆšπ“ d𝓍 using midpoint Riemann sums with n = 20, 50, and 100. A midpoint Riemann sum approximates the 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: Divide the interval [1, 4] into n subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (4 - 1) / n. For n = 20, 50, and 100, compute Δ𝓍 for each case.
Step 3: Determine the midpoints of each subinterval. For each subinterval [𝓍ᡒ, π“α΅’β‚Šβ‚], the midpoint is given by 𝓂ᡒ = (𝓍ᡒ + π“α΅’β‚Šβ‚) / 2. Calculate the midpoints for all subintervals for n = 20, 50, and 100.
Step 4: Evaluate the function 2βˆšπ“ at each midpoint. For each midpoint 𝓂ᡒ, compute f(𝓂ᡒ) = 2βˆšπ“‚α΅’. This gives the height of the rectangle for each subinterval.
Step 5: Compute the midpoint Riemann sum for each value of n. Multiply the function value at each midpoint by the width of the subinterval, Δ𝓍, and sum these products for all subintervals. This gives the approximate value of the integral for n = 20, 50, and 100.

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Key Concepts

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

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The notation βˆ«β‚α΅‡ f(x) dx indicates the integral of f(x) from a to b, providing a numerical value that reflects the accumulation of quantities.
Recommended video:
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Definition of the Definite Integral

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles, calculated using sample points within each subinterval, provides an estimate of the integral. The accuracy of the approximation improves as the number of rectangles (n) increases, making it essential to understand how to compute these sums effectively.
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Introduction to Riemann Sums

Midpoint Rule

The Midpoint Rule is a specific type of Riemann sum where the height of each rectangle is determined by the function value at the midpoint of each subinterval. This method often yields better approximations than using left or right endpoints, especially for functions that are continuous and smooth. By applying this rule with varying values of n, one can observe how the approximation converges to the actual value of the integral.
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Left, Right, & Midpoint Riemann Sums
Related Practice
Textbook Question

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

Ζ’(𝓍) = tan⁻¹ (3x - 1) on [0,1]

(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.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ’, Ζ’', and Ζ’'' are continuous functions for all real numbers.                                                                                                                                                           

                                                                                                                                                                    

(b) βˆ« (Ζ’(𝓍))ⁿ Ζ’'(𝓍) d𝓍 = 1/(n + 1) (Ζ’(𝓍))ⁿ⁺¹ + C , n β‰  ―1 .

Textbook Question

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

I = βˆ«β‚€^Ο€/2 (cos ΞΈ ― 2 sin ΞΈ) dΞΈ = ―1

(b) βˆ«β‚€^Ο€/2 (4 cos ΞΈ ― 8 sin ΞΈ) dΞΈ

Textbook Question

Substitutions Suppose Ζ’ is an even function with βˆ«β‚€βΈ Ζ’(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       

(b) βˆ«Β²β‚‹β‚‚ π“Β²Ζ’(𝓍³) d𝓍

Textbook Question

Area functions for constant functions Consider the following functions Ζ’ and real numbers a (see figure).

(b) Verify that .A'(𝓍) = Ζ’(𝓍)

                                                                                                                                                            

Ζ’(t) = 5 , a = -5

Textbook Question

Generalizing the Mean Value Theorem for Integrals Suppose Ζ’ and g are continuous on [a, b] and let h(𝓍) = (𝓍―b) βˆ«β‚Λ£ Ζ’(t) dt + (𝓍―a) βˆ«β‚“α΅‡g(t)dt.                                                                                                                                                                                                                                                                                                                                

(b) Show that there is a number c in (a, b) such that βˆ«β‚αΆœ Ζ’(t) dt = Ζ’(c) (b ― c)                                                                                                              

                                                                                                                                                                                

(Source: The College Mathematics Journal, 33, 5, Nov 2002)