Skip to main content
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 5.2.25b
Chapter 5, Problem 5.2.25b

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

Verified step by step guidance
1
Step 1: Understand the problem. You are tasked with approximating the net area bounded by the graph of ƒ(𝓍) = tan⁻¹(3x - 1) and the x-axis on the interval [0,1] using Riemann sums (left, right, and midpoint) with n = 4 subintervals.
Step 2: Divide the interval [0,1] into n = 4 subintervals. The width of each subinterval, Δx, is calculated as Δx = (1 - 0)/4 = 1/4.
Step 3: Determine the x-values for the left endpoints, right endpoints, and midpoints of the subintervals. For left endpoints: x₀ = 0, x₁ = 1/4, x₂ = 1/2, x₃ = 3/4. For right endpoints: x₁ = 1/4, x₂ = 1/2, x₃ = 3/4, x₄ = 1. For midpoints: x₀ = 1/8, x₁ = 3/8, x₂ = 5/8, x₃ = 7/8.
Step 4: Evaluate the function ƒ(𝓍) = tan⁻¹(3x - 1) at each of the x-values determined in Step 3. For example, for left endpoints, calculate ƒ(0), ƒ(1/4), ƒ(1/2), and ƒ(3/4). Repeat this for right endpoints and midpoints.
Step 5: Compute the Riemann sums. Multiply each function value by the width of the subinterval, Δx = 1/4, and sum them up. For the left Riemann sum: S_left = Δx * [ƒ(0) + ƒ(1/4) + ƒ(1/2) + ƒ(3/4)]. For the right Riemann sum: S_right = Δx * [ƒ(1/4) + ƒ(1/2) + ƒ(3/4) + ƒ(1)]. For the midpoint Riemann sum: S_mid = Δx * [ƒ(1/8) + ƒ(3/8) + ƒ(5/8) + ƒ(7/8)].

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
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 total area under a curve by dividing the interval into smaller subintervals. Each subinterval's area is estimated using the function's value at specific points, such as the left endpoint, right endpoint, or midpoint. The sum of these areas provides an approximation of the net area under the curve over the specified interval.
Recommended video:
06:11
Introduction to Riemann Sums

Left, Right, and Midpoint Sums

In Riemann sums, the left sum uses the function's value at the left endpoint of each subinterval, while the right sum uses the value at the right endpoint. The midpoint sum, on the other hand, takes the function's value at the midpoint of each subinterval. Each method yields different approximations of the area, with the midpoint sum often providing a more accurate estimate due to its consideration of the function's behavior within the interval.
Recommended video:
07:39
Left, Right, & Midpoint Riemann Sums

Net Area

The net area refers to the total area between the graph of a function and the x-axis over a specified interval, accounting for both positive and negative areas. Positive areas are above the x-axis, while negative areas are below it. When calculating net area using Riemann sums, it is essential to consider the sign of the function's values to ensure that the total area reflects the correct contributions from both regions.
Recommended video:
05:06
Finding Area When Bounds Are Not Given
Related Practice
Textbook Question

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.


(c) ∫₄⁰ 6𝓍(4 ― 𝓍) d(𝓍)

Textbook Question

Working with area functions Consider the function ƒ and its graph.

(c) Sketch a graph of A, for 0 ≤ 𝓍 ≤ 10 , without a scale on the y-axis.


Textbook Question

{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.


∫₁⁴ 2√𝓍 d𝓍

Textbook Question

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

                                                                                                                                                                    

(b) ∫ (ƒ(𝓍))ⁿ ƒ'(𝓍) d𝓍 = 1/(n + 1) (ƒ(𝓍))ⁿ⁺¹ + C , n ≠ ―1 .

Textbook Question

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

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(b) ∫₀^π/2 (4 cos θ ― 8 sin θ) dθ

Textbook Question

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

                                                                                                                                                                    

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .