# 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 $$A + B + C + D + F$$.

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