How to find the volume of cross-sectional area of a cylinder

plane S: 2x-y+2z=2

cylinder: x^2+y^2=1 (z>0)

I am not good with matlab. I started using it a month ago.

I want to find the volume of the area and center of volume below the cross sectional area of the cylinder and plane S.(z>0) without integration.

(Well, in this case with specific numbers, I might be able to deal with using simple integrals.

But, since this is part of a physics project I want to change the factor of the plane. That, I can't integrate because it gets too complicated)

So, is there a function or a way to find volume and center of mass of an area that I can express it with inequality?

[Torsten]

Your question is unclear. What do you mean by

"volume of the area and center of volume below the cross sectional area of the cylinder and plane S.(z>0) "

You mean the volume of the cylinder under the plane and above the x/y plane ? And the center of gravity of this volume ?

[서준 장]

Yes. you're right. it is the 'volume of the cylinder under the plane and above the x/y plane', and the center of gravity of it. Sorry for the unclear expressions.

Answer 1

[John D'Errico]

Is there a ready made function that will do the mathematics for you, based only on an inequality? No. You will need to do some mathematics. Or at least, you will need to do some thinking. You could use integral2, to compute the volume. I can think of at least two ways to do that, depending on if you want to work in polar coordinates or not.

Or, you can use symmetry considerations, and find the point where the plane crosses the z axis, thus at x=y=0. (HINT: Look at the equation of the plane. When x=y=0, what must z be?) Now, think about the volume of stuff in the cylinder that lies above and below that point in z. You can now compute the volume of the simple cylinder with that height. Do you know the formula for the volume of a simple cylinder? Look it up. The simple horizontally topped cylinder has the same volume as the cylinder with the sloping top. (If you don't understand this, then you will just need to do the mathematics I mentioned above. Sorry, but this is far too much homework with no effort shown by you for me to do any more on this question.)

[서준 장]

Thank you so much for you're answer. I think I considered this problem so difficult. I think I can solve this with integrel 2 like what you said.

Thank you!