On this page we discuss integration and surface area of parametric equations.
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To calculate the surface area defined by revolving a parametric curve defined as \( x=X(t)\) and \(y=Y(t)\) from \(t=t_0\) to \(t=t_1\) about the xaxis, we use this integral.
\(\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{ Y(t) \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~ dt}}\)
If you compare this integral to the equation for arc length (in the prevous section) you will see the common factor
\( \sqrt{[X'(t)]^2 + [Y'(t)]^2} \)
Some textbooks write the surface integral differently, taking this into account. You may see it written as
\(\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{Y(t) ~ ds}}\)
where \( ds = \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~dt \)
When rotating about the yaxis, the integral we use is
\(\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{X(t) ~ ds}}\)
Notice we use a capital S to represent surface area. This is the standard symbol you will see in many textbooks. We reserve a lowercase s to represent arc length.
Practice
Find the surface area of revolution of the parametric curve \(x = t^4+4\), \(y = 8t\), \( 0 \leq t \leq 2 \) rotated about the xaxis.
Problem Statement
Find the surface area of revolution of the parametric curve \(x = t^4+4\), \(y = 8t\), \( 0 \leq t \leq 2 \) rotated about the xaxis.
Solution
video by The Organic Chemistry Tutor 

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Find the surface area of revolution of the parametric curve \(x = t^2\), \(y = t^3\), \( 0 \leq t \leq 1 \) rotated about the yaxis.
Problem Statement
Find the surface area of revolution of the parametric curve \(x = t^2\), \(y = t^3\), \( 0 \leq t \leq 1 \) rotated about the yaxis.
Solution
video by The Organic Chemistry Tutor 

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Find the surface area of revolution of the parametric curve \(x = 3t^2\), \(y = 2t^3\), \(0 \leq t \leq 5\) rotated about the yaxis.
Problem Statement 

Find the surface area of revolution of the parametric curve \(x = 3t^2\), \(y = 2t^3\), \(0 \leq t \leq 5\) rotated about the yaxis.
Final Answer 

\(\displaystyle{ S_y = \frac{24\pi}{5} \left[ 949\sqrt{26}+1 \right] }\)
Problem Statement
Find the surface area of revolution of the parametric curve \(x = 3t^2\), \(y = 2t^3\), \(0 \leq t \leq 5\) rotated about the yaxis.
Solution
video by Krista King Math 

Final Answer
\(\displaystyle{ S_y = \frac{24\pi}{5} \left[ 949\sqrt{26}+1 \right] }\)
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Find the surface area of revolution of the parametric curve \(x = 1t\), \(y = 2\sqrt{t}\), \(1 \leq t \leq 4\) rotated about the xaxis.
Problem Statement
Find the surface area of revolution of the parametric curve \(x = 1t\), \(y = 2\sqrt{t}\), \(1 \leq t \leq 4\) rotated about the xaxis.
Solution
video by Krista King Math 

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