Question

-(x² + y² + z²)

Use spherical coordinates to evaluate the triple integral ∭ ᴇ e / √x² + y² + z² dV,
where E' is the region bounded by the spheres x² + y² + z² = 36 and x² + y² + z² = 16

Answer 1

The student is asked to evaluate a triple integral using spherical coordinates in mathematics, specifically calculating the integral over the region bounded by two spheres with radii 4 and 6.

Step-by-step explanation:

The student's question involves evaluating a triple integral using spherical coordinates. The integral is to be evaluated over the region between two spheres with radii 4 and 6. In spherical coordinates, the volume element dV is expressed as ρ^2 sin(φ) dρ dθ dφ, where ρ is the radial distance, θ is the azimuthal angle, and φ is the polar angle.

The integral is set up as an iterated integral with bounds for ρ from 4 to 6, for θ from 0 to 2π, and for φ from 0 to π. The function to be integrated, e / √(x² + y² + z²), simplifies to e / ρ in spherical coordinates, making the integral ∫∫∫ e / ρ • ρ^2 sin(φ) dρ dθ dφ. We perform the integration step by step by first integrating with respect to ρ, then θ, and finally φ.