   Chapter 15.5, Problem 12E

Chapter
Section
Textbook Problem

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 determine

To find: The area of given surface.

Explanation

Given:

The function is the part of the sphere, x2+y2+z2=4z.

The region D lies inside the paraboloid z=x2+y2.

Formula used:

The surface area with equation z=f(x,y),(x,y)D, where fx and fy are continuous, is A(S)=D[fx(x,y)]2+[fy(x,y)]2+1dA.

Here, D is the given region.

If f is a polar rectangle R given by 0arb,αθβ where 0βα2π, then,Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

Calculation:

Solve the given equations.

x2+y2+z2=4zz+z2=4zz23z=0z(z3)=0

This yields, z=0 or z=3. This z=3 intersects the given function. Therefore,

x2+y2+z2=4zx2+y2+z24z+44=0x2+y2+(z2)2=4x2+y2+(32)2=4

Simplify further as follows.

x2+y2+(1)2=4x2+y2+1=4x2+y2=41x2+y2=3

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 3 and θ varies from 0 to 2π. The given equation becomes,

x2+y2+(z2)2=4(z2)2=4x2y2z2=4x2y2z=2+4x2y2

Obtain the partial derivatives of f with respect to x and y

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