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Calculus

Calculus of Parametric Equations

Parametric Equations - Area

         

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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