Question

Consider the following integral: ∫∫∫ e⁽ˣ² ⁺ ʸ²⁾ dv, where e lies between the spheres x² + y² + z² = 9 and x² + y² + z² = 16. Write an equivalent integral using spherical coordinates.

Answer 1

The equivalent integral using spherical coordinates is ∫∫∫ e⁽ˣ² ⁺ ʸ²⁾ r² sin(θ) dr dθ dφ, where 0 ≤ r ≤ 4, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π.

Step-by-step explanation:

To write the given integral using spherical coordinates, we need to express the differential volume element dv in terms of the spherical coordinates. The differential volume element in spherical coordinates is given by dv = r² sin(θ) dr dθ dφ.

Considering the given spheres, the radius of the inner sphere is 3 (√9) and the radius of the outer sphere is 4 (√16).

Therefore, the equivalent integral using spherical coordinates is: ∫∫∫ e⁽ˣ² ⁺ ʸ²⁾ r² sin(θ) dr dθ dφ, where 0 ≤ r ≤ 4, 0 ≤ θ ≤ π, and 0 ≤ φ ≤ 2π.