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 5.1.31d

Chapter 5, Problem 5.1.31d

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6
(d) Calculate the left and right Riemann sums.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with calculating the left and right Riemann sums for the function ƒ(𝓍) = e^(𝓍/2) over the interval [1, 4] with n = 6 subintervals. Riemann sums approximate the area under a curve by summing the areas of rectangles.

Step 2: Divide the interval [1, 4] into n = 6 subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (4 - 1) / 6 = 3 / 6 = 0.5.

Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate the function. The left endpoints are: 𝓍₀ = 1, 𝓍₁ = 1.5, 𝓍₂ = 2, 𝓍₃ = 2.5, 𝓍₄ = 3, and 𝓍₅ = 3.5. Compute ƒ(𝓍) = e^(𝓍/2) at each of these points.

Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate the function. The right endpoints are: 𝓍₁ = 1.5, 𝓍₂ = 2, 𝓍₃ = 2.5, 𝓍₄ = 3, 𝓍₅ = 3.5, and 𝓍₆ = 4. Compute ƒ(𝓍) = e^(𝓍/2) at each of these points.

Step 5: Multiply each function value by the width of the subinterval, Δ𝓍 = 0.5, and sum the results for both the left and right Riemann sums. The left Riemann sum is Σ[ƒ(𝓍ᵢ) * Δ𝓍] for i = 0 to 5, and the right Riemann sum is Σ[ƒ(𝓍ᵢ) * Δ𝓍] for i = 1 to 6.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 left Riemann sum uses the left endpoints of the subintervals, while the right Riemann sum uses the right endpoints.

Definite Integral

The definite integral of a function over an interval represents the net area under the curve of the function between two points. It is a fundamental concept in calculus that connects the concept of accumulation with the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral is denoted as ∫[a,b] f(x) dx, where [a,b] is the interval of integration.

Function Evaluation

Function evaluation involves substituting specific values into a function to determine its output. In the context of Riemann sums, evaluating the function at the endpoints of the subintervals is crucial for calculating the sum. For the given function f(x) = e^(x/2), evaluating it at the left and right endpoints of each subinterval will provide the necessary function values to compute the left and right Riemann sums.

Recommended video:

4:26

Evaluating Composed Functions

Related Practice

Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.

(e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)

views

Textbook Question

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.

views

Textbook Question

Sigma notation Evaluate the following expressions.

(e) 3

∑ (2m + 2) / 3

m =1

views

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.

views

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.

∫₀^π/2 cos 𝓍 d𝓍 ; n = 4

views

Textbook Question

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

ƒ(𝓍) = 1/x on [1,6] ; n = 5

(d) Calculate the midpoint Riemann sum.

235

views

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6(d) Calculate the left and right Riemann sums.

Verified video answer for a similar problem:

Key Concepts

Riemann Sums

Definite Integral

Function Evaluation

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.
{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6
(d) Calculate the left and right Riemann sums.