The top edge of a steel support beam in a theme-park ride is modeled by the cubic function y=g(x)=41x3−2x, where x and y are measured in feet. The beam runs from x=0 to x=4.
(a) Write a definite integral that represents the exact length of the top edge of the beam. Do not attempt to simplify the integrand.
(b) Evaluate the integral numerically to the nearest tenth of a foot.
(c) A straight beam connecting (0,g(0)) to (4,g(4)) would be easier to manufacture. Compute the length of this straight beam and determine, to the nearest tenth of a foot, how much longer the modeled curved beam is than the straight beam.