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Calculus

Calculus of Parametric Equations

Concept Quizzes

Parametric Equations - Derivative
Parametric Equations - Tangent
Parametric Equations - Area
Parametric Equations - Arc Length
Parametric Equations - Velocity and Acceleration
Parametric Equations - Surface Area
Parametric Equations - Volume

Challenge Quizzes

Parametric Equations Calculus: Level 3 Challenges

Parametric Equations - Surface Area

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{1+e^2}+\ln(e+\sqrt{1+e^2})-\sqrt{2}- \ln(\sqrt{2}+1)\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{2+e^2}+\ln(e+\sqrt{2+e^2})-\sqrt{2}- \ln(\sqrt{2}+2)\right)}

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by Brilliant Staff

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

\displaystyle{ \left(\frac{2}{3} - \frac{17\sqrt{272} }{6}\right)\pi}

\displaystyle{ \left(\frac{256}{387} - \frac{1088\sqrt{272} }{389}\right)\pi}

\displaystyle{ \left(\frac{32}{49} - \frac{272\sqrt{272} }{97}\right)\pi}

\displaystyle{ \left(\frac{256}{385} - \frac{544\sqrt{272} }{197}\right)\pi}

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by Brilliant Staff

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

16 \sqrt{20} \pi

8 \sqrt{52} \pi

16 \sqrt{52} \pi

8 \sqrt{20} \pi

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by Brilliant Staff

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

196 \pi

49 \pi

98 \pi

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by Brilliant Staff

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

51 \pi

48 \pi

45 \pi

43 \pi

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by Brilliant Staff