Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 58

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 areaA = 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)