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.
97. ∫ (from 0 to 1) tan(x²) dx; n = 40

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with estimating the integral ∫ (from 0 to 1) tan(x²) dx using three numerical methods: the Midpoint Rule, the Trapezoidal Rule, and Simpson’s Rule, with n = 40 subintervals.
2. Step 2: Divide the interval [0, 1] into n = 40 subintervals. The width of each subinterval, denoted as Δx, is calculated as Δx = (b - a) / n, where a = 0 and b = 1. Thus, Δx = 1 / 40.
3. Step 3: For the Midpoint Rule M(n), calculate the midpoint of each subinterval. The midpoint for the i-th subinterval is given by xᵢ = a + (i - 0.5)Δx. Evaluate the function tan(x²) at each midpoint and sum the results, then multiply by Δx to approximate the integral.
4. Step 4: For the Trapezoidal Rule T(n), calculate the function values at the endpoints of each subinterval. The formula for the Trapezoidal Rule is T(n) = (Δx / 2) * [f(a) + 2Σf(xᵢ) + f(b)], where xᵢ are the interior points of the subintervals. Compute the sum and multiply by Δx / 2.
5. Step 5: For Simpson’s Rule S(n), use the formula S(n) = (Δx / 3) * [f(a) + 4Σf(xᵢₒ) + 2Σf(xᵢₑ) + f(b)], where xᵢₒ are the odd-indexed points and xᵢₑ are the even-indexed points within the interval. Compute the sums, apply the weights (4 for odd-indexed points, 2 for even-indexed points), and multiply by Δx / 3 to approximate the integral.

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 methods that estimate the area under a curve by using discrete data points. 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 subdivisions used.

Midpoint Rule

The Midpoint Rule is a numerical integration technique that approximates the integral of a function by evaluating the function at the midpoint of each subinterval. For a given interval divided into n equal parts, the area under the curve is estimated by summing the areas of rectangles whose heights are determined by the function values at 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 Rule and Trapezoidal Rule to achieve higher accuracy. By using quadratic approximations, Simpson's Rule can yield significantly better results than linear methods, particularly for functions that exhibit curvature.