The curve C is defined for by the parametric equations x(t)=t
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
The curve C is defined for 0≤t≤T by the parametric equations
x(t)=t,y(t)=ln(sect).
A particle begins at the point corresponding to t=0 and travels along C as t increases.
(1) Derive an integral expression for the distance L(T) the particle has traveled by the time it reaches parameter value T. Your answer should be written as a definite integral with explicit limits.
(2) Evaluate the integral obtained in part (a) to obtain an explicit formula for L(T) in terms of T. Simplify completely.
(3) Given that L(T)=2, determine the value of T to three decimal places.
(4) Using your answer to part (c), give the coordinates of the particle when it has traveled exactly 2 units along the curve.