EXAMPLE 3 (a) Set up the integral for the length of the arc of the hyperbola xy = 2 from the point (1, 2) to the point 3, 2 3 . (b) Use Simpson's Rule with n = 10 to estimate the arc length. SOLUTION (a) We have y = 2 x dy dx = and so the arc length is L = 3 1 + dy dx 2 dx 1 = 3 1 + 4 x4 dx 1 = 3 x4 + 4 x2 dx 1 . (b) Using Simpson's Rule with a = 1, b = 3, n = 10, Δx = 0.2, and f(x) = , we have L = 3 1 + 4 x4 dx 1 ≈ Δx 3 [f(1) + 4f(1.2) + 2f(1.4) + 4f(1.6) + ⋯ + 2f(2.6) + 4f(2.8) + f(3)] ≈ (rounded to four decimal places).