95–98. {Use of Tech} Numerical integration Estimate the following integrals using the Midpoint Rule M(n), the Trapezoidal Rule T(n), and Simpson’s Rule S(n) for the given values of n.
96. ∫ (from 1 to 3) dx/(x³ + x + 1); n = 4

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with estimating the integral ∫ (from 1 to 3) dx/(x³ + x + 1) using three numerical methods: the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule, with n = 4 subintervals.
2. Step 2: Divide the interval [1, 3] into n = 4 subintervals. The width of each subinterval, denoted as Δx, is calculated as Δx = (b - a)/n, where a = 1 and b = 3. Compute Δx.
3. Step 3: Apply the Midpoint Rule. For each subinterval, find the midpoint, evaluate the function f(x) = 1/(x³ + x + 1) at each midpoint, and multiply the sum of these values by Δx. The formula for the Midpoint Rule is M(n) = Δx * Σ f(midpoints).
4. Step 4: Apply the Trapezoidal Rule. Evaluate the function f(x) = 1/(x³ + x + 1) at the endpoints of each subinterval and compute the average height of the trapezoids. The formula for the Trapezoidal Rule is T(n) = (Δx/2) * [f(x₀) + 2Σf(xᵢ) + f(xₙ)], where x₀ and xₙ are the endpoints, and xᵢ are the intermediate points.
5. Step 5: Apply Simpson's Rule. Use the formula S(n) = (Δx/3) * [f(x₀) + 4Σf(odd xᵢ) + 2Σf(even xᵢ) + f(xₙ)], where x₀ and xₙ are the endpoints, and odd/even xᵢ refer to the intermediate points. This method requires n to be even, which is satisfied here (n = 4).

Key Concepts

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. It involves using discrete data points to estimate the area under a curve. Common methods include the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule, each with varying degrees of accuracy depending on the function and the number of intervals used.

Midpoint Rule

The Midpoint Rule is a numerical integration technique that approximates the area under a curve by dividing the interval into equal subintervals and using the midpoint of each subinterval to calculate the function's value. The area is then estimated as the sum of the areas of rectangles formed by these midpoints. This method can provide a better approximation than using the endpoints, especially for functions that are relatively smooth.

Simpson's Rule

Simpson's Rule is a more advanced numerical integration method that approximates the integral of a function by fitting parabolas to segments of the curve. It requires an even number of subintervals and combines the Midpoint and Trapezoidal Rules to achieve higher accuracy. By using quadratic approximations, Simpson's Rule can yield significantly better results than simpler methods, particularly for functions that exhibit curvature.