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 5.1.33

Chapter 5, Problem 5.1.33

A midpoint Riemann sum Approximate the area of the region bounded by the graph of ƒ(𝓍) = 100 ― x² and the x-axis on [0, 10] with n = 5 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure).

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Step 1: Divide the interval [0, 10] into n = 5 subintervals. The width of each subinterval, Δx, is calculated as Δx = (10 - 0) / 5 = 2.

Step 2: Determine the midpoints of each subinterval. The subintervals are [0, 2], [2, 4], [4, 6], [6, 8], and [8, 10]. The midpoints are x₁ = 1, x₂ = 3, x₃ = 5, x₄ = 7, and x₅ = 9.

Step 3: Evaluate the function ƒ(x) = 100 - x² at each midpoint to find the height of each rectangle. For example, ƒ(1) = 100 - 1², ƒ(3) = 100 - 3², and so on.

Step 4: Calculate the area of each rectangle using the formula Area = height × width. The width of each rectangle is Δx = 2, and the height is given by ƒ(midpoint). For example, Area₁ = ƒ(1) × 2, Area₂ = ƒ(3) × 2, and so on.

Step 5: Add up the areas of all rectangles to approximate the total area under the curve. The total area is Σ (ƒ(midpoint) × Δx) for all subintervals.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Riemann Sum

A Riemann sum is a method for approximating the total area under a curve by dividing the region into smaller subintervals and summing the areas of rectangles formed. The height of each rectangle can be determined using different points within the subintervals, such as the left endpoint, right endpoint, or midpoint. In this case, the midpoint Riemann sum uses the midpoint of each subinterval to calculate the height, providing a more accurate approximation of the area.

Midpoint Rule

The midpoint rule is a specific type of Riemann sum where the height of each rectangle is determined by the function value at the midpoint of each subinterval. This approach often yields better approximations of the area under a curve compared to using endpoints, as it tends to balance the overestimation and underestimation of the area, especially for curves that are not linear.

Definite Integral

A definite integral represents the exact area under a curve between two specified points on the x-axis. It is the limit of Riemann sums as the number of subintervals approaches infinity, providing a precise calculation of the area. In this context, the definite integral of the function ƒ(x) = 100 - x² from 0 to 10 would give the exact area bounded by the curve and the x-axis, which the midpoint Riemann sum approximates.

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ƒ(𝓍) = 𝓍³ on [―1, 1]

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