The function `y = sqrt(r^2 - x^2) ` describes a circle centred on the origin with radius `r ` .
If we revolve this function in the range `0 <=x <=a `, `a < r ` about the y-axis we obtain a surface of revolution that is specifically a zone of a sphere with radius `r `.
A zone of a sphere is the surface area between two heights on the sphere (surface area of ground between two latitudes when thinking in terms of planet Earth).
For the range of interest `0 <=x<=a `, the zone of interest is specifically a spherical cap on the sphere of radius `r `. The range of interest for `y ` corresponding for that for ` ``x ` is `sqrt(r^2-a^2) <= y <= r ` .
The equivalent on planet Earth of the surface area of such a spherical cap could be, for example, the surface area of a polar region. This of course makes the simplifying assumption that the Earth is perfectly spherical, which is not the case.
To calculate the surface area of this cap of a sphere with radius `r ` , we require the formula for the surface area of revolution of a function `x = f(y) ` (note, I have swapped the roles of `x ` and `y ` for convenience, as the formula is typically written for rotating about the x-axis rather than about the y-axis as we are doing here).
The formula for the surface area of revolution of a function `x = f(y) ` rotated about the y-axis in the range `alpha <= y <= beta ` is given by
`A = int_alpha^beta 2pi x sqrt(1+ ((dx)/(dy))^2) \quad dy`
Here, we have that `alpha = sqrt(r^2 - a^2) ` and `beta = r ` . Also, we have that
` (dx)/(dy) = -y/sqrt(r^2-y^2)`
so that the cap of interest has area`A = int_sqrt(r^2-a^2)^r 2pi sqrt(r^2-y^2) sqrt(1+(y^2)/(r^2-y^2)) \quad dy`
which can be simplified to
`A =2pi int_sqrt(r^2-a^2)^r sqrt((r^2-y^2) + y^2) \quad dy`
`= 2pi int_sqrt(r^2-a^2)^r r dy ` `= 2pi r y |_sqrt(r^2-a^2)^r = 2pi r (r -sqrt(r^2-a^2)) `
So that the zone (specifically cap of a sphere) area of interest A =
`= pi (2r^2 - 2rsqrt(r^2-a^2)) ` ` `
This marries up with the formula for the surface area of a spherical cap
`A = pi (h^2 + a^2) `
where `a ` is the radius at the base of the spherical cap and `h ` is the height of the cap. The value of `h `is the range covered on the y-axis, so that
`h = r -sqrt(r^2 - a^2)` and
`h^2 = 2r^2 - 2rsqrt(r^2 - a^2) - a^2 ` and
`h^2 + a^2 = 2r^2 - 2rsqrt(r^2 - a^2) `