What are the Limits for Computing the Volume of a Cylinder Inside a Sphere?

[boneill3]

Hi Guys,

I have been given the coordinates of a cylinder inside a sphere and want to convert to Cylindrical coordinates to compute the volume of the cylinder.

Can you please check the limits and integral I have?

The cylinder is x^2+y^2= 4

sphere = x^2+y^2+z^2= 9

As its a cylinder we have

Limits are 0<= theta <= 2\pi 0<= r <= 2 and

Inside a sphere with limits

sphere = x^2+y^2+z^2= 9

z = sqrt{9-r^2}

So would my integral be:


\int{{0}{2\pi} \int{0}{2} \int{0}{sqrt{9-r^2}} r dz dr d(theta)


regards

 

[arildno]

Why should the lower limit of z be 0?

Inside a sphere with limits

sphere = x^2+y^2+z^2= 9

z = sqrt{9-r^2}

Are you sure about z=sqrt(9-r^2)) is the only limit set upon z by the above equation?

 

[boneill3]

Sorry I want to compute the solid bounded above and below by the sphere and inside the cylinder.

I see your point the sphere can be either side of the z axis .

it should be:

int{-sqrt{9-r^2}} {sqrt{9-r^2}} r d(theta)


Is that alright

 

[arildno]

That's right indeed.

 

[boneill3]

Thanks for your help!