# Find the area of the zone of a sphere formed by revolving the graph of y=sqrt(9-x^2) , 0<=x<=2 about the y-axis.

mathsworkmusic | Certified Educator

The function y = sqrt(9 -x^2)   describes a circle centred on the origin with radius 3.

If we rotate this function in the range 0 <=x <=2   about the y-axis we obtain a surface of revolution that is specifically a zone of a sphere with radius 3.

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).

When 0 <=x<=2   for this given sphere (which can be written equivalently as x^2 + y^2 = 9 ) the corresponding range for y  is sqrt(5) <=y <= 3

Since this range includes the top of the sphere, the zone we are considering is more specifically a cap. The equivalent on planet Earth would be a polar region.

To calculate the surface area of this cap of a sphere with radius 3, 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 a <=y <=b  is given by

A = int_a^b 2pi x sqrt(1+ ((dx)/(dy))^2) \quad dy

Here, we have that a = sqrt(5)  and b = 3  . Also, we have that

(dx)/(dy) = -y/sqrt(9-y^2)

so that the cap of interest has area

A = int_sqrt(5)^3 2pi sqrt(9-y^2) sqrt(1+(y^2)/(9-y^2)) \quad dy

which can be simplified to

A = 2pi int_sqrt(5)^3 sqrt((9-y^2) + y^2) \quad dy  = 2pi int_sqrt(5)^3 3 dy = 6pi y |_sqrt(5)^3

So that the zone (specifically cap of a sphere) area of interest A = pi(18 - 6sqrt(5))

This can also be calculated using the formula for calculating the surface area of the cap of a sphere as A = pi (a^2 + h^2)  where a is the radius at the lower limit of the cap and h is the perpendiculat height of the cap. Here this would give A = pi (4 + (3-sqrt(5))^2) = pi (4 + 9 -6sqrt(5) + 5) = pi(18 -6sqrt(5))  (ie the same result).