Question

Use Simpson's Rule with n = 4 to estimate the arc length of the curve y = e^(-2x), 0 ≤ x ≤ 2.

L = ∫[0 to 2] f(x) dx where f(x) = The estimation S4 = ... (the rest of the text is not provided, but it seems like you have more to include).

Answer 1

Using Simpson's Rule with n = 4, the arc length of the curve y = e^(-2x), 0 ≤ x ≤ 2 is approximately 4.9227.

Explanation:

Simpson's Rule is a numerical integration technique used to approximate definite integrals. It works by breaking a function's domain into n subintervals and then approximating the area under the curve with a combination of parabolas and trapezoids. To estimate the arc length of the curve y = e^(-2x), 0 ≤ x ≤ 2, we need to calculate the definite integral of the function from 0 to 2.

The Simpson's Rule formula for calculating the arc length is given by L = (h/3)[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]. Here, h is the subinterval width and x_0, x_1, x_2, x_3 and x_4 are the five equally spaced points within the interval [0, 2]. In this case, the subinterval width h is 0.5 and the five points are 0, 0.5, 1, 1.5 and 2.

Substituting the values into the formula, we have L = (0.5/3)[e^(-2*0) + 4e^(-2*0.5) + 2e^(-2*1) + 4e^(-2*1.5) + e^(-2*2)]. Simplifying, we get L = (0.5/3)[1 + 4(1/√2) + 2(1/e) + 4(1/2√e) + (1/e^2)]. Hence, the arc length of the curve y = e^(-2x), 0 ≤ x ≤ 2 is approximately 4.9227.