Calculus of Parametric Equations

Parametric Equations - Arc Length


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?

What is the length of the curve parametrized by the equations x=e2tcost,y=e2tsint,\begin{array}{c}\displaystyle x=e^{2t}\cos t, & y=e^{2t}\sin t,\end{array} in the domain 0t4?0 \leq t \leq 4 ?

If x=4sin2tx=4\sin^2 t and y=4cos2t,y=4\cos^2 t, what is the distance traveled by the point P=(x,y)P=(x,y) during the time interval 0t5π?0 \leq t \leq 5\pi?

Given the curve H(t)=23(t+4)3/2 H(t) = \frac{2}{3} (t+4)^{3/2} , the arc length of the graph between t=4t=4 and t=12t=12 can be expressed in the form abcd \frac {a\sqrt{b}}{c} - d where aa, bb, cc, and dd are positive integers, aa and cc are coprime, and bb is not divisible by the square of any prime. What is a+b+c+da+b+c+d?

Given the curve defined by x=t3x = t^3 and y=t2,y = t^2, what is the length of the curve from t=0t=0 to t=10?t= 10?


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