Use polar coordinates to find the volume of the given solid.
22. Inside the sphere x2 + y2 + z2 = 16 and outside the cylinder x2 + y2 = 4
To find: The volume of the given solid by using polar coordinates.
The region D lies inside the sphere and outside the cylinder.
The spherical surface is, .
The cylindrical region is, .
If is a polar rectangle R given by where , then, (1)
If is the function of and is the function of then,
From the given region D, it is observed that the sphere and the cylinder intersects the xy-plane. So, the value of r varies from 2 to 4 and the value of varies from 0 to .
Substitute in the equation (1) and obtain the required volume.
Integrate the function with respect to r and by using the equation (2).
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