(Approximating the surface area of an ellipsoid) A physical geographer studying the flow of streams around oval stones needed to calculate the surface areas of many such stones that he modelled as ellipsoids: \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } + \frac { z ^ { 2 } } { c ^ { 2 } } = 1\) . He wanted a simple formula for the surface area so that he could implement it in a spreadsheet containing the measure- ments \(a , b\) , and \(c\) of the stones. Unfortunately, there is no exact formula for the area of a general ellipsoid in terms of ele- mentary functions. However, there are such formulas for ellip- soids of revolution, where two of the three semi-axes are equal. These ellipsoids are called spheroids; an oblate spheroid (like the earth) has its two longer semi-axes equal; a prolate spheroid (like an American football) has its two shorter semi-axes equal. A reasonable approximation to the area of a general ellipsoid can be obtained by linear interpolation between these two. To be specific, assume the semi-axes are arranged in de- creasing order \(a \geq b \geq c\) , and let the surface area be \(S ( a , b , c\) ).

(a) Calculate \(S ( a , a , c\) ), the area of an oblate spheroid.

(b) Calculate \(S ( a , c , c\) ), the area of a prolate spheroid.

(c) Construct an approximation for \(S ( a , b , c\) ) that divides the interval from \(S ( a , a , c\) ) to \(S ( a , c , c\) in the same ratio that \(b\) divides the interval from \(a\) to \(c\) .

(d) Approximate the area of the ellipsoid \(x ^ { 2 }\)