AP Calculus BC 9.3 Finding Arc Lengths of Curves - Exam Style Questions - MCQs - New Syllabus
Question
What is the length of the curve defined by the parametric equations \(x(t)=t^{3}+t\) and \(y(t)=t^{2}-6t+1\) from \(t=0\) to \(t=4\)?
(A) \(78.063\)
(B) \(72.217\)
(C) \(68.469\)
(D) \(49.041\)
(B) \(72.217\)
(C) \(68.469\)
(D) \(49.041\)
▶️ Answer/Explanation
Correct answer: (B) \(72.217\)
Arc length (parametric): \[ L \;=\; \int_{0}^{4}\! \sqrt{\Big(\tfrac{dx}{dt}\Big)^{2}+\Big(\tfrac{dy}{dt}\Big)^{2}}\;dt \;=\; \int_{0}^{4}\! \sqrt{(3t^{2}+1)^{2}+(2t-6)^{2}}\;dt. \] Numerically, \[ L \;\approx\; 72.217. \]