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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.

Area - Basic

         

What is the area between the \(x\)-axis and the parametric curve \[x=6\cos \theta, y=7\sin \theta\] in the interval \(0 \leq \theta \leq \pi?\)

What is the area of the region bounded by the \(x\)-axis and the parametric curve \[\begin{array} & x=t^2 & y=3+18t-5t^2 \end{array}\] in the interval \( 0 \le t \le 4?\)

What is the area between the \(x\)-axis and the parametric curve defined by \(x=t, y=-t^2+11?\)

What is the area between the \(x\)-axis and the parametric curve defined by \(x=5t^3\) and \(y=e^t\) in the interval \(0 \leq t \leq 3?\)

Let \(a, b, k \) be positive integers such that \( 0 < a < b < k \). Consider the points \(P=(-a, ka)\) and \(Q=(b, kb) \) which lie on the graph of \(y=k\lvert x \rvert \). If the area of the region bounded by \(y=k\lvert x \rvert \) and the line segment \(PQ\) is \(95\), what is \(k\)?

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