Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.2.31d

Chapter 5, Problem 5.2.31d

{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.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Verified step by step guidance

Step 1: Understand the problem. The integral ∫₃⁶ (1 - 2𝓍) d𝓍 represents the area under the curve of the function f(𝓍) = 1 - 2𝓍 from x = 3 to x = 6. We are tasked with determining which Riemann sum (left or right) underestimates or overestimates the value of the definite integral, given n = 6 subintervals.

Step 2: Divide the interval [3, 6] into n = 6 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (b - a) / n, where a = 3 and b = 6. Substitute the values to find Δ𝓍.

Step 3: For the left Riemann sum, use the left endpoints of each subinterval to approximate the integral. The formula for the left Riemann sum is: Sₗ = Σ [f(𝓍ᵢ) * Δ𝓍], where 𝓍ᵢ represents the left endpoint of each subinterval. Compute the left endpoints and set up the summation.

Step 4: For the right Riemann sum, use the right endpoints of each subinterval to approximate the integral. The formula for the right Riemann sum is: Sᵣ = Σ [f(𝓍ᵢ) * Δ𝓍], where 𝓍ᵢ represents the right endpoint of each subinterval. Compute the right endpoints and set up the summation.

Step 5: Analyze the behavior of the function f(𝓍) = 1 - 2𝓍. Since f(𝓍) is a decreasing linear function, the left Riemann sum will underestimate the integral (as it uses lower values of f(𝓍)), while the right Riemann sum will overestimate the integral (as it uses higher values of f(𝓍)).

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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 ∫ₐᵇ f(x) dx indicates the integral of f(x) from a to b, providing a numerical value that reflects the net area between the curve and the x-axis.

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by dividing the area under the curve into smaller rectangles. The sum can be calculated using left endpoints, right endpoints, or midpoints of the subintervals. The choice of endpoints affects whether the sum underestimates or overestimates the integral, depending on the function's behavior over the interval.

Underestimation and Overestimation

In the context of Riemann sums, underestimation occurs when the chosen rectangles fall below the curve, while overestimation happens when they extend above it. For a decreasing function, the left Riemann sum will typically overestimate the integral, and the right sum will underestimate it. Conversely, for an increasing function, the left sum will underestimate, and the right sum will overestimate the integral.

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Related Practice

Textbook Question

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

(d) If ∫ₐᵇ ƒ(𝓍) d𝓍 = ∫ₐᵇ ƒ(𝓍) d𝓍, then ƒ is a constant function.

Textbook Question

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

(d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then

B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(8)

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) ∫₀³ 5ƒ(𝓍) d𝓍

Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4

(d) Calculate the left and right Riemann sums.

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Textbook Question

Sigma notation Evaluate the following expressions.

(d) 5

∑ (1 + n²)

n=1