thanks for any help.
I'm trying to find the surface area of a cone via integration.
I know that the parametric equation of a cone is $$x=u\cos(p) \\ y=u\sin(p) \\ z=u$$
So as a vector, $\vec{R} = \langle u\cos(p), u \sin(p), u \rangle$.
Since the area equals the double integral of $ds$, and $d\vec{s} = \dfrac{d\vec{R}}{du} du \times \dfrac{d\vec{R}}{dp} dp$, I work out that:
$$\vec{ds} = u\,du \, dp \, \langle -\cos(p),-\sin(p),1\rangle \\ ds = n \, d\vec{s} \\ ds = u \, du \, dp $$
I would expect I'd get the correct answer if I integrated this between the limits 0 to $2\pi$ and 0 to $h$, however I get $h^2\pi$ which is incorrect. Could someone point me to where I'm going wrong?
Best Answer
You actually had just about everything right, except that you skipped an important step: your normal vector to the surface $ \ \vec{ds} \ $ is correct, but you need to integrate its length over the surface of the cone nappe in order to obtain the surface area.
I'll generalize the problem a little, since the choice of proportions for the cone hides one of the factors in the surface area result. For a cone nappe with a height $ \ h \ $ and a "base radius" $ \ r \ $ , we can use similar triangles to find the parametrization (using your notation)
$$ x \ = \ \left( \frac{r}{h} \right) u \ \cos \ p \ \ , \ \ y \ = \ \left( \frac{r}{h} \right) u \ \sin \ p \ \ , \ \ z \ = \ u \ \ , $$
with the domain $ \ 0 \ \le \ u \ \le \ h \ , \ 0 \ \le \ p \ < \ 2 \pi \ $ . An "upward" normal vector is then given by
$$ \vec{R_u} \ \times \ \vec{R_p} \ \ " = " \ \ \left|\begin{array}{ccc}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ \left( \frac{r}{h} \right) \cos \ p&\left( \frac{r}{h} \right) \sin \ p\quad&1\\ -\left( \frac{r}{h} \right) u \ \sin \ p&\left( \frac{r}{h} \right) u \ \cos \ p\quad&0\end{array}\right| $$
$$ = \ \langle \ -\left( \frac{r}{h} \right) u \ \cos \ p \ \ , \ \ -\left( \frac{r}{h} \right) u \ \sin \ p \ \ , \ \ \left( \frac{r}{h} \right)^2 u \ \rangle \ \ . $$
So, up to this point, your procedure is fine. What is needed now is the "norm" of this vector:
$$ \| \ \vec{R_u} \ \times \ \vec{R_p} \ \| \ \ = \ \ \left[ \ \left( \frac{r}{h} \right)^2 u^2 \ \cos^2 \ p \ + \ \left( \frac{r}{h} \right)^2 u^2 \ \sin^2 \ p \ + \ \left( \frac{r}{h} \right)^4 u^2 \ \right]^{1/2} \ \ . $$
$$ = \ \ \left[ \ \left( \frac{r}{h} \right)^2 u^2 \ + \ \left( \frac{r}{h} \right)^4 u^2 \ \right]^{1/2} \ = \ \left(\frac{r}{h} \right) \ \sqrt{ 1 \ + \ \left( \frac{r}{h} \right)^2 } \ \ u \ \ . $$
It is the "magnitude" of the infinitesimal patches associated with the normal vectors that we wish to integrate over the domain of the parameters. Thus,
$$ S \ \ = \ \ \int_0^{2 \pi} \int_0^h \ \left(\frac{r}{h} \right) \ \sqrt{ 1 \ + \ \left( \frac{r}{h} \right)^2 } \ \ u \ \ du \ dp $$
$$ = \ \ \left(\frac{r}{h} \right) \ \sqrt{ 1 \ + \ \left( \frac{r}{h} \right)^2 } \ \int_0^{2 \pi} dp \ \int_0^h \ u \ \ du $$
$$ = \ \left(\frac{r}{h} \right) \ \sqrt{ 1 \ + \ \left( \frac{r}{h} \right)^2 } \ \cdot \ 2 \pi \ \cdot \ \left(\frac{1}{2}u^2 \right) \vert_0^h \ \ = \ \left(\frac{r}{h} \right) \ \sqrt{ 1 \ + \ \left( \frac{r}{h} \right)^2 } \ \cdot \ \pi \ h^2 $$
$$ = \ \pi \ h \ \cdot \ \left(\frac{r}{h} \right) \ \cdot \ h \ \sqrt{ 1 \ + \ \left( \frac{r}{h} \right)^2 } \ = \ \pi \ r \ \sqrt{ r^2 \ + \ h^2 } \ \ , $$
or $ \ \pi \ $ times the "base radius" times the "slant height" of the cone nappe, as the surface area is frequently expressed. In your use of the "standard cone", for which $ \ r \ = \ h \ $ , this formula gives us $ \ S \ = \ \pi \ \sqrt{2} \ h^2 \ $ , as you will find for your calculations, with the restoration of the omitted step.