Understanding why we compute surface area as we do

[JD_PM]

Homework Statement

Homework Equations

The Attempt at a Solution

[/B]
The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure.

Question:

1) Why do we differentiate with respect to z to get dS?

I'd say because in the constraining figure (in this case a cylinder which lies on the xy plane), while we move through the figure, we're changing $z$ coordinate.
I guess that if the cylinder were to lie on the, say, yz axis we'd differentiate with respect to x.

Am I right?

 

[LCKurtz]

Homework Statement

View attachment 240507

Homework Equations

View attachment 240508

The Attempt at a Solution

[/B]
The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure.

View attachment 240509
View attachment 240510
Question:

1) Why do we differentiate with respect to z to get dS?

I'd say because in the constraining figure (in this case a cylinder which lies on the xy plane), while we move through the figure, we're changing $z$ coordinate.
I guess that if the cylinder were to lie on the, say, yz axis we'd differentiate with respect to x.

Am I right?

I wouldn't put it that way. As you move along the surface you are changing x and y also, so just because z is changing is not the reason. The sphere $x^2 + y^2 + z^2 = 4a^2$ defines any of its variables implicitly as functions of the other two variables. So you could consider $y$ as a function of $x$ and $z$, $x$ as a function of $y$ and $z$, or $z$ as a function of $x$ and $y$. The reason to choose which one is usually given by looking at the particular problem and the shape and ease of working with the domain of the independent variables. In this problem, the $x,y$ domain is a semicircle whose equation is basically given in the problem so that is a reasonable choice. Whether to use rectangular coordinates in the first place is preferable is another question. I might have more to say about that later today if I have time.