Surface Integral of cylinder.

[hhhmortal]

Homework Statement

I’m trying to integrate the surface of a cylinder.
I know when integrating the surface of a cylinder the surface element is:
ρdØdz

Where ρ² + z² = r²


And for a sphere it is:
r²sinθdθdØ


In a sphere r=ρ

But in a cylinder when I’m integrating its surface, could it be written as:



(r² - z² )½ . dθ.dz

For example ∫∫zdS over a cylinder from 0<z<5 and 0<Ø<2π

Would it be:

∫∫z. (r² - z² )½ . dθ.dz ?

 

[HallsofIvy]

You seem to be confusing "$\rho$" and "r". In spherical coordinates, $\rho$ is the straight line distance from the origin to a point. In cylindrical coordinates, "r" is the same as in polar coordinates- the straight line distance from the origin to a point in the xy-plane. In three dimensional cylindrical coordinates r is the distance from the origin to t he point (x,y,0) directly "below" the point (x,y,z). The "differential of area", on the surface of a sphere of (fixed) radius r, is $r d\theta dz$ where r is a constant. I can see no reason to introduce "$(\rho^2- z^2)^{1/2}$".

 

[Architeuthis Dux]

I would like to clarify that the surface integral of a cylinder is a mathematical operation used to calculate the total surface area of a cylinder. It is represented by the symbol ∫∫dS and is defined as the double integral of the surface element dS over the surface of the cylinder. The surface element for a cylinder is given by ρdØdz, where ρ is the distance from the axis of the cylinder to a point on the surface, Ø is the angle around the axis, and dz is the infinitesimal height of the cylinder.

I understand that you are trying to integrate the surface of a cylinder and have correctly identified the surface element. However, the equation you have provided for the surface element of a sphere is not entirely accurate. The correct equation for the surface element of a sphere is r²sinθdθdØ, where r is the radius of the sphere and θ is the angle from the polar axis. This equation can also be written as ρ²sinθdθdØ, where ρ is the distance from the center of the sphere to a point on the surface.

To answer your question, yes, the surface integral for a cylinder can be written as ∫∫z(r²-z²)½dθdz. This equation is correct for calculating the surface area of a cylinder. However, it is important to note that this equation is specific to the given limits of integration (0<z<5 and 0<Ø<2π). If the limits were different, the equation would also change accordingly.

I hope this clarifies the concept of surface integral of a cylinder for you. Remember, as a scientist, it is important to always use accurate equations and understand the limitations and variations of mathematical operations in different scenarios.