Surface area of a spherical cap by integration

[Moana]

hi guys,
i have a question.
i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap.
why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface?

and then next, what role does 'ds' play in here ?
and how do you know that Sy= 2pi (∫cd) x ds?im really confused and appreciate your answers

 

[SteamKing]

hi guys,
i have a question.
i saw this picture, and i don't really understand how they derived with the formula. The aim is basically to find the formula for the surface area of a spherical cap.
why do you differentiate the x=sqrt(rˆ2-yˆ2)? how does that help to find the surface?

and then next, what role does 'ds' play in here ?
and how do you know that Sy= 2pi (∫cd) x ds?im really confused and appreciate your answers

It looks like the surface area of the cap is being calculated using the arc length of part of the semi-circle whose altitude is h in the upper hemisphere.

Since the equation of the sphere is x² + y² + z² = r², where r is the radius of the sphere, a cross section of the sphere in the x-y plane will be a circle having the equation x² + y² = r². The lower edge of the cap is located at y = r - h below the top of this circle on the y-axis. The x location corresponding to this is ##x = \sqrt{r^2-y^2}##. Once you know x and y, you can calculate the length of a small segment of the circumference of this circle, ds. The rest of the calculation is an application of Pappus' Centroid Theorem:

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

Here, ds has a centroid which is located at x units from the y-axis, which also serves as the axis of revolution for this calculation.