# Surface of the cone inside the cylinder

• Nov 26th 2010, 04:56 PM
Metrika
Surface of the cone inside the cylinder
Help me compose the surface integral to solve this problem, please (Blush)

Calculate the area of the surface of the cone $\displaystyle x^2-y^2-z^2=0$, located inside the cylinder $\displaystyle x^2+y^2=1$.
Make a drawing.
• Nov 26th 2010, 05:22 PM
DeMath
Quote:

Originally Posted by Metrika
Help me compose the surface integral to solve this problem, please (Blush)

Calculate the area of the surface of the cone $\displaystyle x^2-y^2-z^2=0$, located inside the cylinder $\displaystyle x^2+y^2=1$.
Make a drawing.

See drawing; but I do not know how to compose the integral to your problem((

http://www.mathhelpforum.com/math-he...rface-cone.png
• Nov 26th 2010, 08:21 PM
Metrika
omg! stunningly beautiful!
In what program this is done?

Is WolframAlpha?
• Nov 27th 2010, 03:39 AM
HallsofIvy
So that we can write this in regular polar cylindrical coordinates, I am going to swap "x" and "z" in the problem:
$\displaystyle z^2- x^2- y^2= 0$ and $\displaystyle z^2+ y^2= 1$.

In cylindrical coordinates, that is $\displaystyle z^2= r^2$ and $\displaystyle z^2+ r^2 sin^2(\theta)= 1$ so the two surfaces intersect where $\displaystyle r^2+ r^2 sin^2(\theta)= 1$ or $\displaystyle r= \pm \sqrt{\frac{1}{1+ sin^2(\theta)}$. Integrate with $\displaystyle \theta$ from 0 to $\displaystyle 2\pi$ and, for each $\displaystyle \theta$, r from 0 to $\displaystyle \frac{1}{\sqrt{1+ sin^2(\theta)}}$.

The "differential of surface area" for the cone is $\displaystyle \sqrt{2} r drd\theta$.
• Nov 27th 2010, 04:21 AM
DeMath
Quote:

Originally Posted by Metrika
omg! stunningly beautiful!
In what program this is done?

Is WolframAlpha?

I did it with the Maple_13; нere's the code for building

A := plot3d([[r*cos(t),r*sin(t),r*sqrt(cos(2*t))],[r*cos(t),r*sin(t),-r*sqrt(cos(2*t))]], t=-(1/4)*Pi .. (1/4)*Pi, r=0 .. 1, numpoints=3000, color="LightBlue", style=surface):
B := plottools[rotate](A,0,0,Pi):
C := plot3d([cos(t),sin(t),z], t=0 .. 2*Pi, z=-1 .. 1, color=pink, style=wireframe, transparency=0.75, numpoints=2000):
plots[display](A,B,C, axes=normal, scaling=constrained, lightmodel=light2, view=[-1.4 .. 1.4, -1.4 .. 1.4, -1.4 .. 1.4], orientation=[70,62]);