Volume of the intersection of two cylinders by polar co-ordinates

[cybermask]

Volume of the intersection of two cylinders by cylinderical co-ordinates

Homework Statement

find Volume of the intersection of two cylinders by cylindrical co-ordinates

The Attempt at a Solution

IN the attached file I found it's 8(a^3)/3
It should be 16 not 8

 

[cybermask]

I know that the mistake may be trivial but can anyone give me any comment!1

 

[Dick]

Why do you think the upper limit of the dz integration is r*cos(theta)? Don't you have z=sqrt(a^2-y^2)=sqrt(a^2-r^2*sin(theta)^2)?

 

[cybermask]

Why do you think the upper limit of the dz integration is r*cos(theta)? Don't you have z=sqrt(a^2-y^2)=sqrt(a^2-r^2*sin(theta)^2)?

But in the first octent
x^2 + y^2 = r^2
y^2 + z^2 = r^2

so z=x=rcos(theta)

Isn't it?

 

[Dick]

That's only true along the curve where the two cylinders intersect. It's not true everywhere on the surface in the first octant.

 

[cybermask]

That's only true along the curve where the two cylinders intersect. It's not true everywhere on the surface in the first octant.

Thanks Thanks Thanks