Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(c) Evaluate the definite integral by taking the limit as n →∞ of the Riemann sum in part (b).

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Step 1: Recall the formula for the Riemann sum approximation of a definite integral. Divide the interval [1, 4] into n subintervals of equal width Δ𝓍 = (4 - 1)/n = 3/n.

Step 2: Identify the sample points within each subinterval. For simplicity, use the right endpoints of each subinterval: 𝓍ᵢ = 1 + iΔ𝓍, where i ranges from 1 to n.

Step 3: Write the Riemann sum for the function f(𝓍) = 3𝓍 - 2 over the interval [1, 4]. The sum is given by Sₙ = Σᵢ₌₁ⁿ f(𝓍ᵢ)Δ𝓍, where f(𝓍ᵢ) = 3𝓍ᵢ - 2 and Δ𝓍 = 3/n.

Step 4: Substitute 𝓍ᵢ = 1 + iΔ𝓍 and Δ𝓍 = 3/n into the Riemann sum. This gives Sₙ = Σᵢ₌₁ⁿ [(3(1 + i(3/n)) - 2)(3/n)]. Simplify the expression inside the summation.

Step 5: Take the limit as n → ∞ of the Riemann sum Sₙ. Use the properties of summation and limits to evaluate the sum, which will yield the value of the definite integral ∫₁⁴ (3𝓍 - 2) d𝓍.

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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 value of a definite integral by dividing the area under a curve into small rectangles. The sum of the areas of these rectangles provides an estimate of the integral. As the number of rectangles increases (n → ∞) and their width decreases, the Riemann sum approaches the exact value of the integral.

Definite Integral

A definite integral represents the signed area under a curve between two specified limits, in this case, from 1 to 4. It is denoted as ∫ₐᵇ f(x) dx, where f(x) is the function being integrated. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval [a, b].

Limit Process

The limit process in calculus involves evaluating the behavior of a function as it approaches a certain point or value. In the context of Riemann sums, taking the limit as n approaches infinity allows us to refine our approximation of the integral, leading to the exact value. This process is fundamental in defining the concept of integration in calculus.

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