Final answer:
The question is about using Simpson's Rule to approximate the value of π with n = 6 subintervals. Simpson's Rule is a numerical integration method that is typically applied to a specific integral representing the function for π. Without the exact function, an approximation for π could be achieved by applying Simpson's Rule to f(x) = √(1-x^2) over the interval [-1,1].
Step-by-step explanation:
The student's question pertains to using Simpson's Rule to approximate the value of π (pi) with n = 6 subintervals. Simpson's Rule is a method for numerical integration that estimates the area under a curve. It requires the number of subintervals (n) to be even, and it is more accurate than the trapezoidal rule for functions that are well approximated by parabolas. The actual application of Simpson's Rule involves summing the function's values at the start and end points, four times the function's value at each odd subinterval, and twice the function's value at each even subinterval (excluding the first and last). These summed values are then multiplied by the width of each subinterval (Δx) divided by 3. The formula for Simpson's Rule is given by:
- S = Δx/3 [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]
The references provided in the question seem unrelated to the formula or process for employing Simpson's Rule. Because the application of Simpson's Rule in this question presupposes a specific integral and function representing π, students must know this information to provide an accurate approximation. In the absence of such a function, one could consider f(x) = √(1-x^2) on the interval [-1,1], which, when integrated, gives half of the area of a circle of radius 1, or π/2. Doubling the result of this integration would provide an approximation for π. To achieve the desired approximation, the student would need to apply Simpson's Rule to this function over the interval [-1,1] with 6 evenly spaced subintervals.