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 5.1.9

Chapter 5, Problem 5.1.9

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the 𝓍-axis by dividing the interval [1, 7] into n = 6 subintervals. Use a left and right Riemann sum to obtain two different approximations.

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First, determine the width of each subinterval by dividing the total interval length by the number of subintervals: \(\Delta x = \frac{7 - 1}{6} = 1\).

Identify the left endpoints of each subinterval for the left Riemann sum: these are \(x = 1, 2, 3, 4, 5, 6\). For each of these points, find the corresponding function values \(f(x)\) from the graph.

Calculate the left Riemann sum by multiplying each function value at the left endpoints by the width \(\Delta x\) and summing them up: \(L = \sum_{i=0}^{5} f(x_i) \Delta x\) where \(x_i\) are the left endpoints.

Identify the right endpoints of each subinterval for the right Riemann sum: these are \(x = 2, 3, 4, 5, 6, 7\). For each of these points, find the corresponding function values \(f(x)\) from the graph.

Calculate the right Riemann sum by multiplying each function value at the right endpoints by the width \(\Delta x\) and summing them up: \(R = \sum_{i=1}^{6} f(x_i) \Delta x\) where \(x_i\) are the right endpoints.

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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 approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles. The height of each rectangle is determined by the function value at specific points, such as the left or right endpoints of each subinterval.

Left and Right Endpoint Approximations

Left and right Riemann sums use the function values at the left or right endpoints of each subinterval to determine rectangle heights. Left sums may underestimate or overestimate the area depending on the function's behavior, while right sums provide a complementary approximation.

Partitioning the Interval

Dividing the interval [1, 7] into n = 6 equal subintervals means each subinterval has width Δx = (7-1)/6 = 1. This partitioning is essential for calculating the Riemann sums, as it defines the base of each rectangle used in the approximation.

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