Final answer:
To find the total mass of the solid region W defined by given conditions, we can calculate the volume of the region using triple integration and then multiply it by the density. The volume is calculated by setting up a triple integral with appropriate limits of integration. Given the density function, we can substitute it into the integral and solve for the total mass.
Step-by-step explanation:
To find the total mass of the solid region W, we need to calculate the volume of the region and then multiply it by the density.
The region W is defined by the conditions: x ≥ 0, y ≥ 0, x² + y² ≤ 4, and x ≤ z ≤ 2 - x.
First, let's find the limits of integration for x, y, and z.
The region W is a solid disk with radius 2 centered at the origin in the xy-plane. The z-coordinate ranges from x to 2 - x. Therefore, the limits of integration are:
x: 0 to 2
y: 0 to √(4 - x²)
z: x to 2 - x
Next, let's set up the triple integral to calculate the volume:
V = ∫02 ∫0√(4 - x²) ∫x2 - x dz dy dx
This triple integral gives us the volume of the solid region W. To find the total mass, we need to multiply the volume by the density:
M = δV
where δ is the density. Given δ(x, y, z) = 6y g/cm³, we can substitute this expression into the integral and solve for the total mass.
The complete question is: Find the total mass of the solid region W defined by x≥0,y≥0,x²+y² ≤4,x≤z≤22−x (in centimeters) assuming a mass density of δ(x,y,z)=6y g/cm³ is: