Answer:
Simpson's rule is a numerical method used to approximate the definite integral of a function. It provides a more accurate estimate compared to simple methods like the trapezoidal rule. To derive Simpson's rule, we'll start by considering the problem of approximating the integral of a function f(x) over a given interval [a, b].
The idea behind Simpson's rule is to approximate the function f(x) by a quadratic polynomial within each subinterval of [a, b]. The integral of this quadratic polynomial can then be easily computed, providing an estimate for the integral of f(x) over that subinterval.
Let's assume we have an even number of subintervals, n, which means we'll have n+1 equally spaced points: a, x₁, x₂, ..., xₙ₊₁ = b. Each subinterval will have a width of h = (b - a) / n.
Now, we want to approximate the integral of f(x) over the interval [a, b] using Simpson's rule. We'll divide the interval into subintervals of width h and consider the integral over each subinterval separately.
Within each subinterval [xᵢ₋₁, xᵢ₊₁], we'll approximate f(x) by a quadratic polynomial passing through three points: (xᵢ₋₁, f(xᵢ₋₁)), (xᵢ, f(xᵢ)), and (xᵢ₊₁, f(xᵢ₊₁)). Let's denote this quadratic polynomial as Pᵢ(x).
The general form of a quadratic polynomial is P(x) = A + Bx + Cx². We can determine the coefficients A, B, and C by solving a system of equations using the three points mentioned earlier.
Using the three points (xᵢ₋₁, f(xᵢ₋₁)), (xᵢ, f(xᵢ)), and (xᵢ₊₁, f(xᵢ₊₁)), we can set up the following system of equations:
f(xᵢ₋₁) = A + Bxᵢ₋₁ + Cxᵢ₋₁²
f(xᵢ) = A + Bxᵢ + Cxᵢ²
f(xᵢ₊₁) = A + Bxᵢ₊₁ + Cxᵢ₊₁²
Solving this system will give us the coefficients A, B, and C, and thus the quadratic polynomial Pᵢ(x).
Next, we'll calculate the integral of Pᵢ(x) over the subinterval [xᵢ₋₁, xᵢ₊₁]. The integral of a quadratic polynomial is straightforward to compute:
∫[xᵢ₋₁, xᵢ₊₁] Pᵢ(x) dx = ∫[xᵢ₋₁, xᵢ₊₁] (A + Bx + Cx²) dx
= A(xᵢ₊₁ - xᵢ₋₁) + (B/2)(xᵢ₊₁² - xᵢ₋₁²) + (C/3)(xᵢ₊₁³ - xᵢ₋₁³)
Summing up the integrals over all the subintervals, we have:
∫[a, b] f(x) dx ≈ ∑[i=1 to n] ∫[xᵢ₋₁, xᵢ₊₁] Pᵢ(x) dx
≈ ∑[i=1 to n] [ A(xᵢ₊₁ - xᵢ₋₁) + (B/2)(xᵢ₊₁² - xᵢ₋₁²) + (C/3)(xᵢ₊₁³ - xᵢ₋₁³) ]
Simplifying this expression, we obtain:
∫[a, b] f(x) dx ≈ h/3 [ f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f(xₙ₋₁) + 4f(xₙ) + f(xₙ₊₁) ]
This is the formula for Simpson's rule. The integral of f(x) over [a, b] is approximated by evaluating the function at the endpoints and the equally spaced points within each subinterval, multiplying them by the corresponding coefficients (4 for odd-indexed points, 2 for even-indexed points, and 1 for the endpoints), and summing them all up. Finally, we multiply the sum by h/3 to get the approximation of the integral.
Explanation: