The volume of the solid is 3578π/3.
Finding the Volume using Cylindrical Coordinates
Step 1: Define the region in cylindrical coordinates.
The solid is bounded by the sphere x^2 + y^2 + z^2 = 16 (radius 4) and the cone z = sqrt(x^2 + y^2).
In cylindrical coordinates, this region can be expressed as:
0 <= theta <= 2π
0 <= r <= 4
sqrt(r^2) <= z <= 16
Step 2: Set up the integral.
The volume element in cylindrical coordinates is: dV = r * dz * dr * dθ
We want to integrate over the defined region to find the total volume:
V = ∫(0 to 2π) ∫(0 to 4) ∫(sqrt(r^2) to 16) r dz dr dθ
Step 3: Evaluate the integral.
The first integration with respect to z is straightforward:
∫(sqrt(r^2) to 16) r dz = r(16 - sqrt(r^2))
The second integration with respect to r requires a substitution:
∫(0 to 4) r^2 (16 - sqrt(r^2)) dr = 256π/3 - 32π
Finally, the last integration is simple:
∫(0 to 2π) 256π/3 - 32π dθ = 3642π/3 - 64π
Step 4: Find the final volume.
V = 3642π/3 - 64π = 3578π/3
Therefore, the volume of the solid is 3578π/3.