Puzzle #8 – Unappealing Puzzle

It’s the most wonderful time of the year, so what better way to celebrate than with a little integration? Here’s a barely veiled pure maths question in festive guise!

The outer surface of a (jingle) bell is described by the revolution of the function

y(x)=\begin{cases}A\cos^2(x^2),\qquad\mid x\mid\leqslant\sqrt{\frac{\pi}{2}}\\{0},\qquad\qquad\qquad\mid x\mid\geqslant\sqrt{\frac{\pi}{2}}\end{cases}

about the y axis where A is a positive constant.

Question 1: Determine the volume of the solid bell in terms of A.

The bells are packaged in a snugly fitting cylindrical box of height A whose radius is the same as that of the bell’s base.

Question 2: Verify that the bell occupies exactly half the volume of the box.

ImageImage
This post comes with an edition to puzzle #7. Take a look before the apocalypse!

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