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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.

A bullet is formed by revolving the area bounded by the the curve \(y = \ln(x)\) from \(x = 1\) to \(x = e\) about the \(x\)-axis.

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 \(i\).

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The beads for a bracelet are spheres of radius of 5mm with a hole of radius 3mm drilled through the center so that they can be threaded together.

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]

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A vase maker decides to construct a vase whose contour is a cubic polynomial according to the following specifications:

\(\bullet\) 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?

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A mathematician is buying a Christmas tree from *Revolutionary* Christmas Trees.

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 \(y\)

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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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