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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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.

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).