F. Determine the length of the parametric curve given by the following parametric equations. x=3sin(t) y=3cos(t) 0<t<2π
Solution
For given parametric equations
L=∫02π(dxdt)2+(dydt)2dt=∫02π(3cos(t))2+(−3sin(t))2dt=L=\int_{0}^{2\pi}{\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt=}\int_{0}^{2\pi}{\sqrt{\left(3cos(t)\right)^2+\left(-3sin(t)\right)^2}dt=}L=∫02π(dtdx)2+(dtdy)2dt=∫02π(3cos(t))2+(−3sin(t))2dt=
=3∫02πcos(t)2+sin(t)2dt=3∫02πdt=6π=3\int_{0}^{2\pi}{\sqrt{{cos(t)}^2+{sin(t)}^2}dt=}3\int_{0}^{2\pi}dt=6\pi=3∫02πcos(t)2+sin(t)2dt=3∫02πdt=6π
Answer
L=6π
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