A ski jump designer models the centerline of a symmetrical ramp by the parabola for
MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A ski jump designer models the centerline of a symmetrical ramp by the parabola y=4x2+1 for −2≤x≤2, where x is measured horizontally in meters and y is measured vertically in meters.
(a) Verify analytically that the ramp profile is symmetric about the vertical axis. State what this symmetry implies for any definite integral used in arc-length computations over [−2,2].
(b) Set up—but do not yet evaluate—an integral expression that gives the total length of the ramp from x=−2 to x=2. Reduce the limits of integration as far as possible by exploiting symmetry.
(c) Evaluate the integral from part (b) exactly to obtain the length of the ramp. Express the result in simplified closed form, including any logarithmic terms that arise.
(d) Briefly describe how the designer could adjust the ramp’s horizontal span while maintaining geometric similarity, and explain qualitatively how this would affect the length formula you derived.