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). (b) Use Simpson's Rule with n = 10 to estimate the arc length. SOLUTION (a) We have y = 2 dy dx and so the arc length is L - /3 1 + dy dx dx 1 4 dx x*+ 4 dx. (b) Using Simpson's Rule with a = 1, b = 3, n = 10, Ax = 0.2, and f(x) = we have L = 1+ 4 dx [f(1) + 4f(1.2) + 2f(1.4) + 4f(1.6) + ... + 2f(2.6) + 4f(2.8) + f(3)] (rounded to four decimal places).EXAMPLE 4 Find the arc length for the curve y = 3x2 - - In(x) taking Po(1, 3) as the starting point. SOLUTION If f(x) = 3x2 - - In(x), then 24 f' ( x ) = 1 + [f ' ( x ) ]2 = 1 + = 1 + 36x2 - 0.5+ = 36x2 + 0.5 + V 1 + [f ( x) ]2 = Thus the arc length is given by s( x ) = / V1 + [f (t)]2 dt dt For instance, the arc length along the curve from (1, 3) to (3, f(3)) is s(3) = In(3) - 3 24 In(3) 24 (rounded to four decimal places).Use the arc length formula to nd the length ufthe curve y = 4x 1r 1 5 x 5 2. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula. Find the exact length of the curve. V=3+6x32 0zx=1