The length of one arch of the curve y = sin x is given by
L = ∫(from 0 to π) √(1 + cos²(x)) dx.
Estimate L by Simpson's Rule with n = 8.

Verified step by step guidance

1. Identify the function to integrate: here, the integrand is \( f(x) = \sqrt{1 + \cos^{2}(x)} \).
2. Determine the interval and number of subintervals: the integral is from \( 0 \) to \( \pi \), and \( n = 8 \) subintervals means the step size is \( h = \frac{\pi - 0}{8} = \frac{\pi}{8} \).
3. Calculate the values of \( f(x) \) at the points \( x_0 = 0, x_1 = h, x_2 = 2h, \ldots, x_8 = 8h = \pi \). For each \( x_i \), compute \( f(x_i) = \sqrt{1 + \cos^{2}(x_i)} \).
4. Apply Simpson's Rule formula: \[ L \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,7} f(x_i) + 2 \sum_{i=2,4,6} f(x_i) + f(x_8) \right] \].
5. Sum the weighted function values according to Simpson's Rule and multiply by \( \frac{h}{3} \) to estimate the length \( L \).

Key Concepts

Arc Length of a Curve

The arc length of a curve y = f(x) from x = a to x = b is given by the integral L = ∫_a^b √(1 + (dy/dx)²) dx. This formula measures the distance along the curve by summing infinitesimal line segments, accounting for changes in both x and y.

Simpson's Rule

Simpson's Rule is a numerical method to approximate definite integrals by dividing the interval into an even number of subintervals and fitting parabolas through the points. It provides an accurate estimate by weighting function values at endpoints, midpoints, and interior points.

Partitioning the Interval (n = 8)

Choosing n = 8 means dividing the interval [0, π] into 8 equal subintervals for applying Simpson's Rule. This partitioning determines the step size and the points at which the integrand is evaluated, affecting the accuracy of the numerical approximation.