Integration extends to the third dimension with the means to compute the volumes of shapes obtained by revolving regions around an axis, such as spheres, toroids, and paraboloids.

It is then shot straight into a very thick wall (i.e. it does not pierce the other side at all) making a closed cylindrical hole until it stops moving. Then the bullet is carefully extracted without affecting the hole at all, leaving an empty hole with a pointy end where the bullet once was.

The length of the entire hole is \(e+1\). If the volume of the hole can be expressed as \[\pi i e\] where \(i\) is a constant, find the value of \(i\).

What is the volume, in mm\(^3\), of material removed from the bead when the hole is drilled?

[Give your answer to 3 decimal places]

\(\bullet\) The vase is 4 feet tall.

\(\bullet\) The vase is 4 feet in diameter at its widest (which occurs 1 foot from the base).

\(\bullet\) The vase is 2 feet in diameter at its narrowest (which occurs 1 foot from the top).

\(\bullet\) The vase has a flat circular bottom.

Rounded to 3 decimal places, what is the volume of the vase in cubic feet?

He sends this model function ( \(x\) in terms of \(y\) ):

\[ x = \begin{cases} 2 - ( \frac{\lfloor y \rfloor }{2} + \{ y \} ) & 0 \leq y < 3\\ \frac{1}{2} & -1 \leq y < 0\\ 0 & \text{otherwise} \end{cases} \]

This is rotated round the \(y\)-axis to create a solid of revolution to model his tree.

He then sends a volume enlargement scale factor \(s\) by which the volume of the solid is multiplied to make the tree the correct size. He wishes to have a final volume of \(640 \pi \).

Find the value of \(s\).

**Note:** The notation {\(y\)} means the fractional part of \(y\)

The volume of the solid formed by revolving the curve

\[y= \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+1}}\]

bounded between \(x = \frac{1}{8}\) and \(x=\frac{9}{16}\) around the \(x\)-axis is in the the form \(A \pi \ln{\frac{B}{C}}.\)

If \(A\) and \(B\) are square-free, what is the value of \(A \times B \times C?\)

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