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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.

Surface Area - Basic

         

What is the surface area \(S\) of the body of revolution obtained by rotating the curve \(y=e^x,\) \(0 \le x \le 1,\) about the \(x-\)axis?

What is the surface area \(S\) of the body of revolution obtained by rotating the parametric curve \[\begin{array} &x = 8 t^2 + 9 &y = -4 t &0 \leq t \leq 1 \end{array}\] about the \(x\)-axis?

What is the surface area \(S\) of the solid of revolution obtained by rotating the parametric curve \[\begin{array} &x = 4 t &y = 6 t &0 \leq t \leq 2 \end{array}\] about the \(y\)-axis?

What is the surface area \(S\) of the body of revolution obtained by rotating the parametric curve \[\begin{array} &x = 7 \cos t &y = 7 \sin t &0 \leq t \leq \pi \end{array}\] about the \(x\)-axis?

If \(S\) is the surface area of the solid obtained by rotating the parametric curve \[\begin{array} &x = 4 \cos^3 t &y = 4 \sin^3 t &0 \leq t \leq \frac{\pi}{2} \end{array}\] about the \(x\)-axis, what is \(\frac{5}{2}S?\)

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