A research team is studying the graph of on the interval as a simplified model of a…
MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A research team is studying the graph of p(x)=e−x2 on the interval [−1,1] as a simplified model of a bell-shaped bridge deck. Because exact antiderivatives of many expressions involving this function are not elementary, they must combine analytical work with numerical estimation.
(a) Find p′(x).
(b) Show that the integrand necessary for calculating the length of the curve y=p(x) over [−1,1] is an even function, and rewrite the arc-length integral so that it involves only the interval [0,1].
(c) Using technology or an appropriate numerical method, approximate the value of the integral from part (b) to three decimal places. State clearly the numerical technique you used.
(d) The team replaces the interval [−1,1] with [−k,k] for a real number k>0. Derive a single integral expression (with reduced limits) whose value would be the length of y=e−x2 on [−k,k]. Explain why the symmetry you used in part (b) continues to hold for any positive k.