- #1
Telemachus
- 835
- 30
[Urgent] Surface area
Hi there, sorry for the hurry, I had a big problem this weekend and I couldn't study. I have my exam tomorrow, and need help with this. I'm finishing with the topic, and I have almost studied all what I had to study, but this exercise resulted problematic.
This is it.
It asks me to calculate the area of the surface [tex]x^2-y^2-z^2=0[/tex] under the conditions [tex]x\geq{0},y\geq{0},z\geq{0},x\leq{1-z}[/tex]
So I've made the parametrization this way:
[tex]\begin{Bmatrix}x=u\\y=u \cos v \\z=u \sin v\end{matrix}[/tex]
And I've found: [tex]||T_u\times{T_v}||=\sqrt[ ]{2}u[/tex]
Then I made the parametrization for the plane:
[tex]\begin{Bmatrix}x=u\\y=v \\z=1-u\end{matrix}[/tex]
Then I made the integral for the surface, well, I've tried:
[tex]\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{1-u}\sqrt[ ]{2}u dudv[/tex]
I think that the surface is actually not bounded.
Actually, when I did the intersection I've found a parabola:
[tex]\sqrt[ ]{y^2+z^2}=1-z\longrightarrow{y^2+2z-1=0}[/tex]
Which I think makes sense, but the surface would be infinite, and I don't know what to do.
Any help will be thanked.
Homework Statement
Hi there, sorry for the hurry, I had a big problem this weekend and I couldn't study. I have my exam tomorrow, and need help with this. I'm finishing with the topic, and I have almost studied all what I had to study, but this exercise resulted problematic.
This is it.
It asks me to calculate the area of the surface [tex]x^2-y^2-z^2=0[/tex] under the conditions [tex]x\geq{0},y\geq{0},z\geq{0},x\leq{1-z}[/tex]
So I've made the parametrization this way:
[tex]\begin{Bmatrix}x=u\\y=u \cos v \\z=u \sin v\end{matrix}[/tex]
And I've found: [tex]||T_u\times{T_v}||=\sqrt[ ]{2}u[/tex]
Then I made the parametrization for the plane:
[tex]\begin{Bmatrix}x=u\\y=v \\z=1-u\end{matrix}[/tex]
Then I made the integral for the surface, well, I've tried:
[tex]\displaystyle\int_{0}^{2\pi}\displaystyle\int_{0}^{1-u}\sqrt[ ]{2}u dudv[/tex]
I think that the surface is actually not bounded.
Actually, when I did the intersection I've found a parabola:
[tex]\sqrt[ ]{y^2+z^2}=1-z\longrightarrow{y^2+2z-1=0}[/tex]
Which I think makes sense, but the surface would be infinite, and I don't know what to do.
Any help will be thanked.