∫∫∫ r dzdrdθ, where 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 4, and 0 ≤ z ≤ √(r^2 / 3).
Solid description:
z = 0: This represents the bottom plane of the solid.
x² + y² = 16: This represents the cylinder with radius 4, centered at the origin.
z = √(1 / 3) (x² + y²): This is a cone that opens upwards, with its vertex at the origin and axis aligned with the z-axis. The slant height of the cone is equal to its radius (4) as √(1 / 3) * 4 = 4.
Cylindrical coordinates:
θ: Angle between the positive x-axis and the radius vector (range: 0 to 2π).
r: Distance from the origin to the point (range: 0 to 4).
z: Height along the z-axis (range: varies depending on the other dimensions).
Integral setup:
We want to integrate over the volume element dV in cylindrical coordinates: dV = r drdθdz.
Bounds:
θ: 0 to 2π to cover the entire cylinder base.
r: 0 to 4 to cover the entire cylinder radius.
z: 0 to √(r^2 / 3) to integrate within the cone and ensure z = 0 at the bottom and z = 4 at the vertex.
Integrand:
The integrand is simply z, as we're interested in the total volume of the solid.
Therefore, the complete triple integral is:
∫∫∫ r dzdrdθ, where 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 4, and 0 ≤ z ≤ √(r^2 / 3).
This integral, when evaluated, will give the total volume of the solid bounded by the cylinder and the cone.