surface area of a cone

itunescape

1. The problem statement, all variables and given/known data
The question is to derive the surface area of a cone.


2. Relevant equations
slant= square root ( r^2 + h^2)
surface area= int int [square root(fx^2 + fy^2 +1) da]
surface area of cone side= pi *r(r^2+h^2)
3d cone formula: z= h/r(squareroot x^2+y^2)


3. The attempt at a solution
by looking at the structure I know that it is the area of the base (circle) + the area of the slant/side, but when I solve for the surface area using double integrals I'm stuck w/ squareroot 2 in the formula. How can I cancel that out?

i calculated fx as hx/rsqareroot(x^2+y^2)
and fy as hy/rsquareroot(x^2+y^2)

i plugged that into the formula for surface area and got: int int [h/r squareroot(2)] r dr d@
it feels like it isn't right and I don't know how to cancel the sqareroot(2) during integration. Can someone hint me in the right direction?

Dick

fx and fy look ok. But when I compute sqrt(1+(fx)^2+(fy)^2) I don't get what you got for the integrand (in particular, no sqrt(2)). Can you tell us how you got that?

itunescape

sure:
int int squareroot [(hx/rsquareroot(x^2+y^2))^2 + (hy/rsquareroot(x^2+y^2))^2 + 1] da
int int squareroot [(h^2*x^2/r^2 (x^2+y^2)) + (h^2*y^2/r^2(x^2+y^2)) + 1] da
int int squareroot [(h^2/r^2) x^2/(x^2+y^2) + y^2/(x^2+y^2) +1] da
the +1 should change into: (x^2+y^2)/(x^2+y^2)
int int squareroot [h^2/r^2 (x^2+ y^2/(x^2+y^2)) + x^2+y^2/(x^2+y^2)] da
int int h/r squareroot(2) r dr d@

does it look right? :/ theres probablt something big that I'm missing but its so hard to see

Dick

You aren't putting enough parentheses in and you are loosing track of what multiplies what.
In this step:
int int squareroot [(h^2/r^2) x^2/(x^2+y^2) + y^2/(x^2+y^2) +1] da
the h^2/r^2 only multiplies the first two terms, not the 1. Combine them first and multiply by h^2/r^2. Then add the 1.