Question

Rotate around the ?-axis the region ? = {(?, ?) ∈ ? 2 : sin ? < ? < π − ?, 0 < ? < π} to obtain the solid Ω. Find its volume and center of mass.

Answer 1

SOLUTION:

The given expression is :

? = {(?, ?) ∈ ? 2 : sin ? < ? < π − ?, 0 < ? < π}

The region is rotated around Z axis so that occupied region will be Symmetric about X-Z plane.

Now,

1) Find Volume of that Region :

the area of small segment assumed in the region as shown in the figure is,dA

that is, , so when this segment is being rotated around Z axis, it will create a ring, which Volume (dV) is,

Now, the full volume of the region can be calculated by integrating small volume withing given range,

------------------------- (ii)

so,   

2) Find the centre of mass of the given region:

The region is rotated around Z-axis so that occupied region will be Symmetric about X-Z plane. hence the centre of mass will lie in the x-z plane only.

The X - coordinate of Centre of mass is:-

so,

Similarly for

so,