## 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 coordi

Question

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.

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2 weeks 2021-09-28T14:50:24+00:00 1 Answer 0

1. Answer: 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 = ∫∫ 2r dθdr

volume = ∫ 2r ∫dθdr

Using the limits for r and θ, we have:

volume = 2π · [ $$(25 – 5^{2} )^{\frac{3}{2} }$$ + ]

volume = 64π mm³

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