A bullet is formed by revolving the area bounded by the the curve from to about the -axis.
然后商店t 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 . If the volume of the hole can be expressed as where is a constant, find the value of .
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 , 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:
The vase is 4 feet tall.
The vase is 4 feet in diameter at its widest (which occurs 1 foot from the base).
The vase is 2 feet in diameter at its narrowest (which occurs 1 foot from the top).
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 fromRevolutionaryChristmas Trees.
He sends this model function ( in terms of ):
This is rotated round the -axis to create a solid of revolution to model his tree.
He then sends a volume enlargement scale factor by which the volume of the solid is multiplied to make the tree the correct size. He wishes to have a final volume of .
Find the value of .
Note:The notation { } means thefractional partof
The volume of the solid formed by revolving the curve
bounded between and around the -axis is in the the form
If and are square-free, what is the value of