Question

Find the length of the curve. Note: You will need to evaluate your integral numerically. Round your answer to one decimal place.
x = cos(4t), y = sin(3t) for 0 ≤ t ≤ 2π

Please help/explain this vector calc problem!

Answer 1

In general, the formula for the length of a formula which is specified through \(x=x(t)\) and \(y=y(t)\) in the interval \(a\lt t\lt b\) is:
\[L=\int^b_a\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\]
So in this case we have:
\[L=\int^{2\pi}_0\sqrt{1+\frac{1}{16}cos^2(4t)+\frac{1}{9}sin^2(3t)}\]

Answer 2

??

\(\dfrac{d}{dt}\cos(4t) = -4\sin(4t)\)

\(\dfrac{d}{dt}\sin(3t) = 3\cos(3t)\)

Answer 3

Thank you both so much! :)