Find the area of the surface.
12. The part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2+ y2
To find: The area of given surface.
The function is the part of the sphere, .
The region D lies inside the paraboloid .
The surface area with equation , where and are continuous, is .
Here, D is the given region.
If is a polar rectangle R given by where , then, (1)
Solve the given equations.
This yields, or . This intersects the given function. Therefore,
Simplify further as follows.
Convert the given problems into the polar coordinates to make the problem easier. By the given the conditions, it is observed that, r varies from 0 to and varies from 0 to . The given equation becomes,
Obtain the partial derivatives of f with respect to x and y
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