Final answer:
To compute the length of the curve of f(x)= ln(x) between x=1 and x=3, we can set up an integral using the arc length formula. Simpson's rule with n=4 can be used to approximate the length, and the expression for the approximation is given. The error of the approximation can be estimated using the error formula and we can find how large n should be chosen for the approximation to be accurate within 0.001. To find the exact length of the curve, we can evaluate the integral using calculus.
Step-by-step explanation:
(a) To set up an integral to compute the length of curve, we can use the arc length formula. The formula to find the length of a curve between two points is given by:
Length = ∫[a,b] √(1 + (f'(x))²) dx
For the function f(x) = ln(x) between x=1 and x=3, the integral becomes:
Length = ∫[1,3] √(1 + (1/x)²) dx
(b) Simpson's rule with n=4 is a numerical method to approximate integrals. The expression for Sₙ(Simpson's rule approximation) is given by:
Sₙ = (h/3) * [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f(xₙ-2) + 4f(xₙ-1) + f(xₙ)]
where h = (b-a)/n is the step size, and x₀, x₁, x₂, ..., xₙ are the equally spaced points between a and(c) The error of the Simpson's rule approximation can be estimated using the error formula:
∣Eₛ(n)∣ ≤ K(b-a)⁵/180n⁴
Given K=25 and the desired accuracy of 0.001, we can solve for n in the inequality:
0.001 ≤ 25(3-1)⁵/180n⁴Simplifying, we get:0.001 ≤ 0.00555555/n⁴
Taking the reciprocal on both sides, we get:
n⁴ ≤ 5555.55Take the fourth root of both sides:
n ≤ 8.35Rounding up to the nearest whole number, we need to choose n ≥ 9 to ensure the approximation Sₙ is accurate within 0.001.
(d) To find the exact length of the curve between x=1 and x=3, we can use the arc length formula:
Length = ∫[1,3] √(1 + (1/x)²) dx
Using calculus, we can evaluate this integral to find the exact length.