Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.


∫₁⁴ (𝓍²―1) d𝓍

Verified step by step guidance

1. Step 1: Recall the definition of the definite integral using Riemann sums. The definite integral ∫ₐᵇ f(x) dx can be approximated by a sum: lim(n→∞) Σᵢ₌₁ⁿ f(xᵢ)Δx, where Δx = (b - a)/n and xᵢ = a + iΔx for right Riemann sums.
2. Step 2: Identify the function f(x) = x² - 1, the interval [1, 4], and the number of subintervals n. Calculate Δx = (4 - 1)/n = 3/n.
3. Step 3: Determine the sample points xᵢ for the right Riemann sum. For i = 1, 2, ..., n, xᵢ = 1 + iΔx = 1 + i(3/n).
4. Step 4: Substitute f(xᵢ) and Δx into the Riemann sum formula. The sum becomes Σᵢ₌₁ⁿ [(xᵢ² - 1) * Δx], where xᵢ = 1 + i(3/n) and Δx = 3/n.
5. Step 5: Simplify the expression for the sum and take the limit as n → ∞ to evaluate the definite integral. This involves expanding (1 + i(3/n))², simplifying the summation, and using summation formulas for i and i².

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫_a^b f(x) dx, where 'a' and 'b' are the limits of integration. The value of the definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into rectangles. The sum of the areas of these rectangles, calculated using sample points (like right endpoints), provides an estimate of the integral. As the number of rectangles increases and their width decreases, the Riemann sum approaches the exact value of the definite integral.

Theorem 5.1 (Fundamental Theorem of Calculus)

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is continuous on [a, b], then the definite integral of its derivative over that interval equals the difference in the values of the function at the endpoints. This theorem provides a powerful tool for evaluating definite integrals and establishes the relationship between the antiderivative and the area under the curve.