Area of Surface of Revolution/Polar Form

Theorem

Let $\SS$ be a surface of revolution such that:

$\SS$ is embedded in a cylindrical coordinate space $\polar {r, \theta, z}$
the axis of revolution of $\SS$ is aligned with the polar axis
the curve $\CC$ being rotated to generate $\SS$ is the plane curve:
$r = \map r \theta$
where $\theta$ is in the closed interval $\closedint a b$.


Then the area of $\SS$ is given by:

$\ds \map \AA \SS = 2 \pi \int_a^b r \sin \theta \sqrt {r^2 + \paren {\dfrac {\d r} {\d \theta} }^2} \rd \theta$



This article needs proofreading.
In particular: the source is imprecise and the actual geometry is not explicitly specified
If you believe all issues are dealt with, please remove {{Proofread}} from the code.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Proofread}} from the code.


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



Sources

  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): area of a surface of revolution
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): area of a surface of revolution
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.