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Calculus of Parametric Equations

Parametric equations can be quite handy, and we don't want to unravel them just to do Calculus. Some tricks can bend traditional derivative and integral methods to apply to 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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