Final answer:
To find the area of the surface that lies inside the paraboloid and the sphere, we need to determine the intersection between the two surfaces. By setting the equations of the paraboloid and the sphere equal to each other and simplifying, we find the equation of the intersection. Then, we use the formula for the surface area of a sphere to find the area of the surface.
Step-by-step explanation:
To find the area of the surface that lies inside the paraboloid and the sphere, we need to determine the intersection between the two surfaces.
First, let's find the intersection equation by setting the equations of the paraboloid and the sphere equal to each other:
x2 + y2 + z2 = 4z and z = x2 + y2
Substituting the second equation into the first equation, we get:
x2 + y2 + (x2 + y2)2 = 4(x2 + y2)
Simplifying this equation, we have:
x2 + y2 + x4 + 2x2y2 + y4 = 4x2 + 4y2
Rearranging terms, we get:
x4 + 2x2y2 + y4 - 3x2 - 3y2 = 0
Now, we use the formula for the surface area of a sphere to find the area of the surface:
Surface area = 4πr2
In this case, the radius of the sphere is 2 because the equation is x2 + y2 + z2 = 4z. So, the surface area of the sphere is 4π(2)2 = 16π.