Calculus of Parametric Equations

Parametric Equations Calculus: Level 3 Challenges


If \(y\) is a function of \(x\) as defined by the parametric relation \(y=3\sin ^{ 2 }{ t }\) when \(x=\tan { t } \), then determine the value of \(\displaystyle\lim_{ x\rightarrow \infty }{ y }\) .

The parametric equation of a cycloid is given below.

\[\large x = a(t - \sin t) \\ \large y = a(1 - \cos t)\]

What is the area of the region bounded by the two arcs of the cycloid in the above figure?

What is the length of the arc of the curve \(x^{\frac{2}{3}} + y^{\frac{2}{3}}=4\)?

The graph above satisfy the equation \(x^4 + y^3 = x^2 y \).

The area enclosed by the 2 cute adorable little fine loops is equals to \( \frac {a}{b} \) for coprime positive integers \(a\) and \(b\). What is the value of \(a+b\)?

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 \(0 \leq t \leq 2\pi\)?


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