Question

Now making a specific bead from a sphere of radius 3 and drilling out a cylinder through the center of radius 1. Which of the following is a correct setup in polar coordinates to determine the volume?

Answer 1

The volume of the bead is found by subtracting the volume of the cylinder drilled out of the sphere from the volume of the original sphere, employing cylindrical shells in polar coordinates and considering the varying radius from the cylinder to the sphere.

Step-by-step explanation:

To determine the volume of a bead created from a sphere of radius 3 and drilled with a cylinder of radius 1, you can employ the method of cylindrical shells in polar coordinates. First, you need to calculate the volume of the cylinder that is removed from the sphere. The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder, and h is the height, which in this case would be the diameter of the sphere (twice the radius of the sphere).

After finding the volume of the cylinder, subtract this from the volume of the original sphere. The volume of a sphere is given by V = 4/3 πr³, where r is the radius of the sphere. Because the cylinder's axis is the same as the sphere's axis, in polar coordinates, we consider the volume elements (shells) as the radius varies from the outer radius of the cylinder (1) to the radius of the sphere (3).