Final answer:
To solve the given parts, one must establish a recursive formula between Iᵢ₊₁ and Iᵢ, and use it for simplifying the evaluation of I₈. Then, apply Simpson's rule with the given function and intervals to approximate π.
Step-by-step explanation:
To answer part (a) of the question, let's consider the expression for I₍₊₁ which involves the integral of (x² + 4) raised to the power (n+1). If we expand this expression in terms of I₍, we can try to identify coefficients A₍ and B₍ such that:
I₍₊₁ = A₍ I₍ + B₍
To do this, look for a relationship between (x² + 4) raised to the n and (n+1) powers that would allow for the integral to be expressed in terms of I₍ directly or through differentiation/integration by parts.
For part (b), you must use the relationship found in part (a) repeatedly to simplify the calculation of the integral with n=8. Basic algebra and calculus techniques will be used to evaluate I₈.
In part
, Simpson's rule approximation of π involves dividing the interval from 0 to 2 into four equal subintervals and then applying the Simpson's rule formula:
π ≈ ¾·[(f(0)+4f(½)+2f(1)+4f(1 ½)+f(2))]
Where f(x) = (x² + 4)/8. Finally, plugging in the values of f(x) at the points x = 0, ½, 1, 1 ½ and 2 will yield the Simpson's rule approximation of π.