159k views
3 votes
Use cylindrical or spherical coordinates, whichever seems more appropriate, to find the volume of the solid that lies above the cone z=sqrt(x² + y²) and below the sphere x² + y² + z² = 9?

1 Answer

6 votes

Final answer:

To calculate the volume of the solid above the cone z=sqrt(x² + y²) and below the sphere x² + y² + z² = 9, spherical coordinates are utilized in a triple integral with bounds for radius ρ from 0 to 3, angle φ from 0 to π/4, and angle θ from 0 to 2π.

Step-by-step explanation:

To find the volume of the solid that lies above the cone z=sqrt(x² + y²) and below the sphere x² + y² + z² = 9, spherical coordinates seem more appropriate due to symmetry. The volume element in spherical coordinates is dV = ρ² sin(φ)dρ dφ dθ. We must integrate over the appropriate bounds for ρ, φ, and θ.

  • The radius ρ goes from the surface of the cone to the surface of the sphere, which translates to ρ going from 0 to 3.
  • The angle φ, which measures the angle from the positive z-axis, will go from 0 to π/4 because the cone has a slope of 1, corresponding to a 45-degree angle.
  • The angle θ goes all the way around the circle, so from 0 to 2π.

Integrating within these bounds using a triple integral will give us the volume of the solid.

User Juude
by
8.8k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories