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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 5.2.21b
Chapter 5, Problem 5.2.21b

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


ƒ(x) = 4 - 2x on [0,4]


(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.

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Step 1: Understand the problem. You are tasked with approximating the net area bounded by the graph of ƒ(x) = 4 - 2x and the x-axis on the interval [0, 4] using three types of Riemann sums: left, right, and midpoint, with n = 4 subintervals.
Step 2: Divide the interval [0, 4] into n = 4 equal subintervals. The width of each subinterval, Δx, is calculated as Δx = (b - a) / n, where a = 0 and b = 4. Substitute the values to find Δx.
Step 3: For the left Riemann sum, use the left endpoints of each subinterval to evaluate ƒ(x). Multiply each ƒ(x) value by Δx and sum them up to approximate the net area.
Step 4: For the right Riemann sum, use the right endpoints of each subinterval to evaluate ƒ(x). Multiply each ƒ(x) value by Δx and sum them up to approximate the net area.
Step 5: For the midpoint Riemann sum, use the midpoints of each subinterval to evaluate ƒ(x). Multiply each ƒ(x) value by Δx and sum them up to approximate the net area.

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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 region 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.
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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, and understanding their differences is crucial for accurate calculations.
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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 given interval, accounting for areas above the x-axis as positive and those below as negative. This concept is essential when calculating the overall area, as it reflects the function's behavior across the interval, particularly when the function crosses the x-axis.
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Finding Area When Bounds Are Not Given
Related Practice
Textbook Question

Sigma notation Evaluate the following expressions.                                                                                                                                          

(b)    10                                                                                                                                                                               

       ∑  (2κ + 1)                                                                                                                                                                          

       κ=1                         

Textbook Question

Displacement from a table of velocities The velocities (in mi/hr) of an automobile moving along a straight highway over a two-hour period are given in the following table.

(b) Find the midpoint Riemann sum approximation to the displacement on [0,2] with n = 2 and .n = 4 .

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Textbook Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(b) Graph ƒ and A.

ƒ(𝓍) = eˣ ; a = 0 , b = ln 2 , c = ln 4

Textbook Question

Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).

(b) Verify that A'(𝓍) = ƒ(𝓍).

ƒ(t) = 3t + 1 , a = 2

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.


∫₀⁴ (4𝓍― 𝓍²) d𝓍

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Textbook Question

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(b) 4 + 5 + 6 + 7 + 8 + 9