Question

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?

Answer 1

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.