Consider the parametric curve where

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the parametric curve

x(t)=t,\quad y(t)=\frac{1}{k}\sin(kt),\quad 0\le t\le \frac{\pi}{k},

where

k>0

. One arch of this curve is traced exactly once as

t

runs over the given interval. Engineers wish to select

k

so that the total length of the arch is

5

units. (a) Write an integral expression

L(k)

that represents the exact length of the arch in terms of

k

. Do not evaluate the integral. (b) Explain briefly why the integral in part (a) cannot be expressed in terms of elementary functions, citing an appropriate theorem or classification of integrals. (c) Use a numerical technique of your choice to approximate the value of

k

that makes

L(k)=5

. Show all set-up work, include at least two iterations or refinements, and report your answer correct to three decimal places. (d) Justify the accuracy of your approximation by analyzing the error bound or convergence properties of the numerical method you used.

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