Answer:
To write an iterated integral representing the mass of the region E with the given mass density we need to integrate the density function `f(x y z) = z^2` over the region E.
First let's find the limits of integration for each variable.
For z we need to find the limits of z-values for which both surfaces intersect. So we set the equations of the surfaces equal to each other:
x^2 + y^2 + z^2 = 4
4x^2 + 4y^2 - z^2 = 1
Rearranging the second equation we have z^2 = 4x^2 + 4y^2 - 1. Substituting this into the first equation we get:
x^2 + y^2 + (4x^2 + 4y^2 - 1) = 4
5x^2 + 5y^2 - 1 = 4
5x^2 + 5y^2 = 5
x^2 + y^2 = 1
This is the equation of a circle in the xy-plane centered at the origin with radius 1. So the limits of z-values for integration will be the square root of 4 - x^2 - y^2 to the negative square root of 4 - x^2 - y^2 which ensures that we are inside the surface x^2 + y^2 + z^2 = 4 but outside the surface 4x^2 + 4y^2 - z^2 = 1.
Now for x and y since E is a circular region in the xy-plane with radius 1 we can use polar coordinates for integration. In polar coordinates x = r * cos(theta) and y = r * sin(theta where r represents the radius and theta represents the angle.
Using polar coordinates we integrate over the circular region E which can be described as 0 ≤ r ≤ 1 and 0 ≤ theta ≤ 2π.
So the iterated integral representing the mass of region E is:
∫∫∫E z^2 dV = ∫₀²∫₀²π∫√(4 - r²)₋√(4 - r²) z² * r dz dtheta dr