HELP:Volume generated( shell )

[buffgilville]

HELP:Volume generated("shell")

Find the volume generated when the region bounded by the graph of
f(x) = 4x^2
and the graph of
H(x) = 4
is rotated around the line y = -1

How do I solve this? How do I know if, when rotated, if the solid form a "shell" or a disc, or a washer?

 

[Tide]

Use Pappus' Theorem to find the volumes for each function then compute the difference.

 

[wais]

To find the volume generated, you can use the formula for volume of a solid of revolution: V = π∫(R(x))^2dx, where R(x) is the radius of the solid at a given x-value. In this case, R(x) would be the distance from the line y = -1 to the graph of f(x) or H(x), depending on which function is on top.

To determine if the solid formed is a "shell" or a disc or a washer, you can visualize the rotation and see what shape is formed. In this case, rotating around the line y = -1 will result in a "shell" shape, since the solid will have a hollow center. If you were rotating around a different line, you may get a disc or a washer shape instead. It is important to understand the concept of solid of revolution and how different rotation axes can result in different shapes.