Calculus of Parametric Equations

Parametric Equations Calculus: Level 3 Challenges


If yy is a function of xx as defined by the parametric relation y=3sin2ty=3\sin ^{ 2 }{ t } when x=tantx=\tan { t } , then determine the value of limxy\displaystyle\lim_{ x\rightarrow \infty }{ y } .

The parametric equation of a cycloid is given below.

x=a(tsint)y=a(1cost)\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 x23+y23=4x^{\frac{2}{3}} + y^{\frac{2}{3}}=4?

The graph above satisfy the equation x4+y3=x2yx^4 + y^3 = x^2 y .

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

The location of a dot PP at a given time tt in the xyxy plane is given by (x,y)=(tsint,1cost)(x,y) = (t - \sin t, 1 - \cos t). What is the distance traveled by PP in the interval 0t2π0 \leq t \leq 2\pi?


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