The ideal cannoli

Calculus Level 5

Here is how to make the ideal mathematical cannoli:

1. Take a unit circle made of thin dough.
2. Wrap it around a cylinder of radius $\frac{1}{\pi}$ so the ends just touch.
3. Bake until crisp and then fill with delicious cream.
4. Scrape off the excess filling using a straight edge held perpendicular to the long axis and touching symmetric points of the edges as you scrape.

What is the volume of the filling in the cannoli?

Note: Top, front, and side views of the cannoli are shown. Since the excess filling has been scraped away, no filling is visible in the side view. Bon appétit!

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