Find the volume of the solid which is bounded by the cylinders

[Benny]

Q. Find the volume of the solid which is bounded by the cylinders x^2 + y^2 = r^2 and y^2 + z^2 = r^2. To me they don't really look like equations of cylinders, more like circles. Would the term "r" be constant in this case? Or would it be a variable? Even if r is a variable, I don't understand why the equations contain its square, rather than just "r" itself. Are the given equations standard equations for a clinder?

I'm just having trouble interpreting the equations at this stage. Help would be apppreciated.

 

[dextercioby]

Yes,as you can see,in the first equation,the "z" variable is free to take any real value.That means that the circle $x^2 +y^2 =r^2$ is free to move along the "z" axis,and thus generating a surface called "right circular cylinder".

The same goes for the other equation.So you've got 2 intersecting right circular cylinders and you need to find the volume.Better make a drawing to find the limits of integration and then choose cylindrical coordinates.

Daniel.

 

[Benny]

Thanks for the help dex. Although, up to the section of my book from which I got this question, cylindrical and spherical coordinates haven't been covered yet. I'll see if I can find another way around this one.

 

[OlderDan]

Q. Find the volume of the solid which is bounded by the cylinders x^2 + y^2 = r^2 and y^2 + z^2 = r^2. To me they don't really look like equations of cylinders, more like circles. Would the term "r" be constant in this case? Or would it be a variable? Even if r is a variable, I don't understand why the equations contain its square, rather than just "r" itself. Are the given equations standard equations for a clinder?

I'm just having trouble interpreting the equations at this stage. Help would be apppreciated.

Here's a picture of your solid

http://mathworld.wolfram.com/SteinmetzSolid.html

plus a lot more, unfortunately. See if you can work it out for yourself once you understand the shape before you just take the solution. r is constant for the integration. Of course the volume depends on r.