- #1
twoflower
- 368
- 0
Hi,
I have this problem:
Compute volume of solid bounded by these planes:
[tex]
z = 1
[/tex]
[tex]
z^2 = x^2 + y^2
[/tex]
When I draw it, it's cone standing on its top in the origin and cut with the [itex]z = 1[/itex] plane.
So after converting to cylindrical coordinates:
[tex]
x = r\cos \phi
[/tex]
[tex]
y = r\sin \phi
[/tex]
[tex]
z = z
[/tex]
[tex]
|J_{f}(r,\phi,z)| = r
[/tex]
I get
[tex]
0 \leq z \leq 1
[/tex]
[tex]
0 \leq \phi \leq 2\pi
[/tex]
[tex]
0 \leq r \leq 1
[/tex]
And
[tex]
V = \iiint_{M}\ dx\ dy\ dz\ =\ \int_{0}^{2\pi}\int_{0}^{1}\int_{0}^{1} r\ dr\ dz\ d\phi
[/tex]
But I got [itex]\pi[/itex] as a result, which is obviously incorrect :(
Can you see where I am doing a mistake?
Thank you!
I have this problem:
Compute volume of solid bounded by these planes:
[tex]
z = 1
[/tex]
[tex]
z^2 = x^2 + y^2
[/tex]
When I draw it, it's cone standing on its top in the origin and cut with the [itex]z = 1[/itex] plane.
So after converting to cylindrical coordinates:
[tex]
x = r\cos \phi
[/tex]
[tex]
y = r\sin \phi
[/tex]
[tex]
z = z
[/tex]
[tex]
|J_{f}(r,\phi,z)| = r
[/tex]
I get
[tex]
0 \leq z \leq 1
[/tex]
[tex]
0 \leq \phi \leq 2\pi
[/tex]
[tex]
0 \leq r \leq 1
[/tex]
And
[tex]
V = \iiint_{M}\ dx\ dy\ dz\ =\ \int_{0}^{2\pi}\int_{0}^{1}\int_{0}^{1} r\ dr\ dz\ d\phi
[/tex]
But I got [itex]\pi[/itex] as a result, which is obviously incorrect :(
Can you see where I am doing a mistake?
Thank you!