# Surface area of a spherical cap by integration

Tags:
1. Jan 28, 2016

### 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

#### Attached Files:

• ###### Screen Shot 2016-01-28 at 20.59.36.png
File size:
88.1 KB
Views:
1,959
2. Jan 28, 2016

### SteamKing

Staff Emeritus
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.