MATHEMATICS · Multi-part · AP · Calculus BC · Applications of Integration · Applications of Integration
Problem
A planar path C consists of two successive segments.
Segment I: x=y2 for 0≤y≤1.
Segment II: y=x for 1≤y≤2.
(a) For Segment I, write a definite integral whose value equals the length of the segment. Evaluate the integral exactly.
(b) For Segment II, first rewrite the segment so that x is expressed explicitly as a differentiable function of y. Then set up and evaluate an integral for the length of this segment.
(c) Determine the total length of the path C by combining your results from parts (a) and (b).
(d) Provide a clear justification for why it is appropriate to set up the two integrals in parts (a) and (b) using different independent variables. Your explanation should reference at least one property of each segment that necessitates the chosen orientation.