{Use of Tech} Approximating definite integrals with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.


∫₀¹ (𝓍² + 1) d𝓍

Verified step by step guidance

1. Understand the problem: The goal is to approximate the definite integral ∫₀¹ (𝓍² + 1) d𝓍 using numerical methods with different values of n (20, 50, and 100). This involves dividing the interval [0, 1] into n subintervals and calculating the sum of areas of rectangles or trapezoids.
2. Set up the formula for numerical integration: For a definite integral ∫ₐᵇ f(𝓍) d𝓍, the interval [a, b] is divided into n subintervals of equal width Δ𝓍 = (b - a) / n. Here, a = 0, b = 1, and f(𝓍) = 𝓍² + 1.
3. Choose a numerical method: Use the midpoint rule, trapezoidal rule, or Simpson's rule. For example, in the midpoint rule, the approximate integral is given by: ∑ᵢ₌₁ⁿ f(𝓍ᵢ)Δ𝓍, where 𝓍ᵢ is the midpoint of each subinterval.
4. Calculate the midpoints and evaluate the function: For each subinterval, calculate the midpoint 𝓍ᵢ = a + (i - 0.5)Δ𝓍, where i ranges from 1 to n. Then, evaluate f(𝓍ᵢ) = (𝓍ᵢ² + 1) for each midpoint.
5. Sum the results and multiply by Δ𝓍: Add up all the values of f(𝓍ᵢ) and multiply the sum by Δ𝓍 to get the approximate value of the integral. Repeat this process for n = 20, 50, and 100, and compare the results to estimate the integral's value.

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 a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.

Riemann Sum

A Riemann sum is a method for approximating the value of a definite integral by dividing the area under the curve into smaller rectangles. The sum of the areas of these rectangles, calculated using sample points within each subinterval, provides an estimate of the integral. As the number of rectangles increases (n → ∞), the Riemann sum approaches the exact value of the definite integral.

Numerical Integration

Numerical integration refers to techniques used to approximate the value of definite integrals when an analytical solution is difficult or impossible to obtain. Common methods include the Trapezoidal Rule and Simpson's Rule, which utilize Riemann sums and weighted averages to improve accuracy. Calculators and software often implement these methods to provide quick estimates for integrals.