A bead is made by drilling a cylindrical hole of radius 1 mm through a sphere of radius 5 mm. Set up a triple integral in cylindrical coordinates representing the volume of the bead. Evaluate the integral.

Volume = 64π mm³

Step-by-step explanation: A triple integral is generally used to calculate the volume of a region. In this case, the integral will be

V = . The limits of the triple integral will be:

1≤r≤5, in which r is the radius of the sphere;

0≤θ≤2π, which θ is the angle of the sphere;

and the limits determined by the bead;

To find the limits of the bead, we use the cylinfrical coordinates. In it, the sphere is represented by the equation

So, the region of the bead will be:

z = ±

- ≤ z ≤

Calculating and substituing:

volume =

volume = ∫∫∫ r dzdθdr

volume = ∫∫ rz dθdr

Using the limits for z:

volume = ∫∫ r·(√25 - r²) + r·(√25 - r²) dθdr

volume = ∫∫ 2rdθdr

volume = ∫ 2r∫dθdr

Using the limits for r and θ, we have:

volume = 2π · [ + ]

volume = 64π mm³

The volume of a bead inside a sphere is 64π mm³