Parametric Equations - Surface Area Practice Problems Online

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?

\(\displaystyle{ \pi \left( e\sqrt{4+e^2}+\ln(e+\sqrt{4+e^2})-\sqrt{2}- \ln(\sqrt{2}+4)\right)} \) \(\displaystyle{ \pi \left( e\sqrt{2+e^2}+\ln(e+\sqrt{2+e^2})-\sqrt{2}- \ln(\sqrt{2}+2)\right)} \) \(\displaystyle{ \pi \left( e\sqrt{3+e^2}+\ln(e+\sqrt{3+e^2})-\sqrt{2}- \ln(\sqrt{2}+3)\right)} \) \(\displaystyle{ \pi \left( e\sqrt{1+e^2}+\ln(e+\sqrt{1+e^2})-\sqrt{2}- \ln(\sqrt{2}+1)\right)} \)

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?

\[\displaystyle{ \left(\frac{2}{3} - \frac{17\sqrt{272} }{6}\right)\pi} \] \[\displaystyle{ \left(\frac{256}{385} - \frac{544\sqrt{272} }{197}\right)\pi} \] \[\displaystyle{ \left(\frac{256}{387} - \frac{1088\sqrt{272} }{389}\right)\pi} \] \[\displaystyle{ \left(\frac{32}{49} - \frac{272\sqrt{272} }{97}\right)\pi} \]

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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Parametric Equations - Surface Area

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