# volume of a sphere

• Nov 1st 2008, 08:54 AM
calculusgeek
volume of a sphere
find the volume of a sphere x^2 +y^2+z^2<and= 1 contained between planes z=1/2 and z=-1/sqrt(2) using cylindrical coordinates and spherical co ordinates

in terms of cylindrical coords i think the z values in teh intergral are teh vales of teh planes but im then findgin didficult in interpretin how to determine the other limits

and also spherical corords im just looking for hints on how to get the limits so i can try have sokme base to start with so i can tryn figure it out!!!

can anyone help me please thank u
• Nov 1st 2008, 03:47 PM
mr fantastic
Quote:

Originally Posted by calculusgeek
find the volume of a sphere x^2 +y^2+z^2<and= 1 contained between planes z=1/2 and z=-1/sqrt(2) using cylindrical coordinates and spherical co ordinates

in terms of cylindrical coords i think the z values in teh intergral are teh vales of teh planes but im then findgin didficult in interpretin how to determine the other limits Mr F says:

$\displaystyle {\color{red}-\frac{1}{\sqrt{2}} \leq z \leq \frac{1}{2}}$

$\displaystyle {\color{red}0 \leq r \leq 1}$

$\displaystyle {\color{red}0 \leq \theta \leq 2 \pi}$.

and also spherical corords im just looking for hints on how to get the limits so i can try have sokme base to start with so i can tryn figure it out!!!

can anyone help me please thank u

For the spherical coordinates, obviously $\displaystyle 0 \leq r \leq 1$ and $\displaystyle 0 \leq \theta \leq 2 \pi$. So you need to get the range of the azimuthal angle.

I suggest looking at the volume side on:

The cross-section is the part of the circle $\displaystyle x^2 + z^2 = 1$ bounded by the lines $\displaystyle z = -\frac{1}{\sqrt{2}}$ and $\displaystyle z = \frac{1}{2}$.

You know the z-coordinates of the points of intersection of the lines and the circle so you can use simple trigonometry on the obvious right-triangles to get the maximum and minimum azimuthal angles.
• Nov 2nd 2008, 04:27 AM
Roland
Quote:

Originally Posted by mr fantastic
For the spherical coordinates, obviously $\displaystyle 0 \leq r \leq 1$ and $\displaystyle 0 \leq \theta \leq 2 \pi$. So you need to get the range of the azimuthal angle.

I suggest looking at the volume side on:

The cross-section is the part of the circle $\displaystyle x^2 + z^2 = 1$ bounded by the lines $\displaystyle z = -\frac{1}{\sqrt{2}}$ and $\displaystyle z = \frac{1}{2}$.

You know the z-coordinates of the points of intersection of the lines and the circle so you can use simple trigonometry on the obvious right-triangles to get the maximum and minimum azimuthal angles.

I get the azimuthal angles to be -pi/6 and pi/4, however as one of these is negative i am not sure if i can apply this?
As one plane is in the negative z plane, surely the angle could be left as a negative...