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Given a cone in $\Bbb R^3$ with base radius R, height H as follows:

enter image description here

Compute its surface area using surface integral. Note that the area of the base is not included.

Hint: Parametrize the cone using cylindrical coordinates.

I have some questions about this task:

There are many variations for cone equations. Which one I need to use for this case and how do I parametrize it to cylindrical coordinates?

Depending on the outcome of the first question, how do I use the equation/cylindrical coordinates to do the surface integration and what are the domains needed for the two integrals?

Nash J.
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Tolunay Yüksel
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  • Please see the link: https://math.stackexchange.com/questions/855831/surface-integral-of-a-right-circular-cone – Nash J. Nov 26 '19 at 20:13
  • I already saw that, but haven't understood it. There the cone is opened in z direction, but this one here is in x direction. Also in the link he uses h/R*r for z. I am confused by the two r's (r and R). – Tolunay Yüksel Nov 26 '19 at 20:21
  • By the link, I meant the answer (https://math.stackexchange.com/a/855880/439920). The direction of the opening of the cone will not change the surface area. You can modify the answer as per your situation, assuming that the cone is lying on the plane $x = H$. By assuming $\alpha$ to be the vertex angle, you'll have $\tan \alpha = R/H$. At a distance of $x$ from the origin, the radius of the base will be $x \tan \alpha$ (as for $x = R$, i.e, at a distance of $H$ from the origin in the $x$-direction, the radius of the base is $H \tan \alpha = R$). – Nash J. Nov 26 '19 at 20:44
  • Is the area of the base included, when I only do the surface integration as in math.stackexchange.com/a/855880/439920? Or do i need to compute and substract it from the solution of the integration in the end? – Tolunay Yüksel Nov 26 '19 at 20:58
  • As per my current understanding, the area of the base is not included in this. To include the area of the base, you need to add an extra term of $\pi R^2$. However, I am very confident about this as it's been a long time since I used multivariate calculus! – Nash J. Nov 26 '19 at 21:04
  • Yeah I might have unterstood it wrong and should add Pi*R^2 – Tolunay Yüksel Nov 26 '19 at 21:08
  • But you question says NOT to include the base! – Nash J. Nov 26 '19 at 21:10
  • As of my understanding now I think that the task wants to tell me that base is not included in the surface area if it is computed by surface integration and that I have to include it myself my adding, right? – Tolunay Yüksel Nov 26 '19 at 21:14
  • I still haven't figured out the parametrized function when the cone is opening in x direction. I know that it doesn't change surface area, but to write down an exact solution I need to know, how the parametrized form would look like in my case. Do i just switch the x and z coordinates from the answer in your link? – Tolunay Yüksel Nov 26 '19 at 21:42
  • Yes, interchanging $x$ and $z$ in the given link should be fine! PS: The answer has been posted by https://math.stackexchange.com/users/160738/user160738 – Nash J. Nov 26 '19 at 21:48

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