A semicircular arch has profile given analytically by the function for
MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A semicircular arch has profile given analytically by the function f(x)=r2−x2 for −r≤x≤r, where r>0 is the radius of the circle.
Answer the following questions about the length of the arch.
(a) Verify that f is an even function and explain the implication this symmetry has for any definite integral taken over the interval [−r,r].
(b) Write—but do not yet evaluate—an integral expression whose value is the exact length of the entire arch from x=−r to x=r. Reduce the limits of integration as much as possible by making full use of the symmetry you established in part (a).
(c) Evaluate the integral found in part (b) to obtain a closed-form expression for the length of the arch in terms of r. Show all steps in your computation.
(d) Interpret your result from part (c) in the context of a full circle of radius r. Explain why your answer is consistent with elementary geometry.