Respuesta :
Answer: Volume = 64π[tex]\sqrt{6}[/tex] 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 = [tex]\int\limits^a_b {\int\limits^a_b {\int\limits^a_b \, dV[/tex]. 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 [tex]r^{2} + z^{2} = 25[/tex]
So, the region of the bead will be:
[tex]r^{2} + z^{2} = 25[/tex]
z = ±[tex]\sqrt{25 - r^{2} }[/tex]
- [tex]\sqrt{25 - r^{2} }[/tex] ≤ z ≤ [tex]\sqrt{25 - r^{2} }[/tex]
Calculating and substituing:
volume = [tex]\int\limits^A_B \, dV[/tex]
volume = ∫∫∫ r dzdθdr
volume = ∫∫ rz dθdr
Using the limits for z:
volume = ∫∫ r·(√25 - r²) + r·(√25 - r²) dθdr
volume = ∫∫ 2r[tex]\sqrt{25-r^{2} }[/tex]dθdr
volume = ∫ 2r[tex]\sqrt{25-r^{2} }[/tex]∫dθdr
Using the limits for r and θ, we have:
volume = 2π · [[tex]\frac{-2}{3}[/tex][tex](25 - 5^{2} )^{\frac{3}{2} }[/tex] + [tex]\frac{2}{3} (25 - 5^{2} )^{\frac{3}{2} }[/tex] ]
volume = 64π[tex]\sqrt{6}[/tex] mm³
The volume of a bead inside a sphere is 64π[tex]\sqrt{6}[/tex] mm³
