Calculus of Parametric Equations

Parametric Equations - Arc Length


The location of a dot \(P\) at a given time \(t\) in the \(xy\) plane is given by \((x,y) = (t - \sin t, 1 - \cos t)\). What is the distance traveled by \(P\) in the interval \(0 \leq t \leq 2\pi\)?

What is the length of the curve parametrized by the equations \[\begin{array} \displaystyle x=e^{2t}\cos t, & y=e^{2t}\sin t,\end{array}\] in the domain \(0 \leq t \leq 4 ?\)

If \(x=4\sin^2 t\) and \(y=4\cos^2 t,\) what is the distance traveled by the point \(P=(x,y)\) during the time interval \(0 \leq t \leq 5\pi?\)

Given the curve \( H(t) = \frac{2}{3} (t+4)^{3/2} \), the arc length of the graph between \(t=4\) and \(t=12\) can be expressed in the form \[ \frac {a\sqrt{b}}{c} - d \] where \(a\), \(b\), \(c\), and \(d\) are positive integers, \(a\) and \(c\) are coprime, and \(b\) is not divisible by the square of any prime. What is \(a+b+c+d\)?

Given the curve defined by \(x = t^3\) and \(y = t^2,\) what is the length of the curve from \(t=0\) to \(t= 10?\)


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