Volume of Revolution

Volume of Revolution: Level 4 Challenges


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

It is then shot straight into a very thick wall (i.e. it does not pierce through 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+1e+1. If the volume of the hole can be expressed as πie,\pi i e, where ii is a constant, find the value of ii.

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 mm3^3, of material removed from the bead when the hole is drilled?

[Give your answer to 3 decimal places]

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?

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

He sends this model function ( xx in terms of yy ):

x={2(y2+{y})0y<3121y<00otherwise 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 yy-axis to create a solid of revolution to model his tree.

He then sends a volume enlargement scale factor ss by which the volume of the solid is multiplied to make the tree the correct size. He wishes to have a final volume of 640π640 \pi .

Find the value of ss.

Note: The notation {yy} means the fractional part of yy

The volume of the solid formed by revolving the curve

y=1x1x+1y= \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x+1}}

bounded between x=18x = \frac{1}{8} and x=916x=\frac{9}{16} around the xx-axis is in the the form AπlnBC.A \pi \ln{\frac{B}{C}}.

If AA and BB are square-free, what is the value of A×B×C?A \times B \times C?


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