# A calculus problem by D G

**Calculus**Level 5

Define an ellipse with major axis \(3\) and minor axis \(2\):

\[{\frac{x^2}{9} + \frac{y^2}{4} = 1}\]

Now construct a second curve of fixed normal distance \(d\) from the ellipse (defining positive \(d\) as pointing away from the origin). What is the average value of the circumference of this curve as \(d\) ranges from \(0\) to \(2 \pi\)? If your answer can be expressed in the form

\[{A \pi^B + C E(-\frac{D}{F})} \]

for positive integers \(A\), \(B\), \(C\), \(D\), \(F\), \(D\) and \(F\) coprime, where \(E(x)\) refers to the complete elliptic integral of the second kind, find \(A + B + C + D + F\).