# Thread: Surface of the cone inside the cylinder

1. ## Surface of the cone inside the cylinder

Help me compose the surface integral to solve this problem, please

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.

2. Originally Posted by Metrika
Help me compose the surface integral to solve this problem, please

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((

3. omg! stunningly beautiful!
In what program this is done?

Is WolframAlpha?

4. 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 $\displaystyle \sqrt{2} r drd\theta$.

5. 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]);