   Chapter 16.6, Problem 42E

Chapter
Section
Textbook Problem

Find the area of the surface.42. The part of the cone z = that lies between the plane y = x and the cylinder y = x2

To determine

To find: The area for surface of the part of the cone z=x2+y2 that lies between the plane y=x and the cylinder y=x2 .

Explanation

Given data:

The equation of the part of the cone is given as follows.

z=x2+y2

The cone lies between the plane y=x and the cylinder y=x2 .

Formula used:

Write the expression to find the surface area of the plane.

A(S)=D1+(zx)2+(zy)2dA (1)

Write the equation of part of the cone as follows.

z=x2+y2 (2)

Calculation of zx :

Take partial derivative for equation (2) with respect to x.

zx=xx2+y2=12x2+y2(2x)=xx2+y2

Calculation of zy :

Take partial derivative for equation (2) with respect to y.

zy=yx2+y2=12x2+y2(2y)=yx2+y2

Calculation of surface area of plane:

Substitute xx2+y2 for zx and yx2+y2 for zy in equation (1),

A(S)=D1+(xx2+y2)2+(yx2+y2)2dA=D1+x2x2+y2+y2x2+y2dA=D1+x2+y2x2+y2dA=D1+1dA

Simplify the expression as follows

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