   Chapter 16.6, Problem 41E

Chapter
Section
Textbook Problem

Find the area of the surface.41. The part of the plane x + 2y + 3z = 1 that lies inside the cylinder x2 + y2 = 3

To determine

To find: The area for the surface of the part of the plane x+2y+3z=1 that lies inside the cylinder x2+y2=3 .

Explanation

Given data:

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

x+2y+3z=1

Rearrange the equation.

3z=1x2yz=1x2y3

z=13x32y3 (1)

The equation of the cylinder is given as follows.

x2+y2=3

Formula used:

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

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

Write the expression to find base surface area of cylinder with radius.

A=πr2 (3)

Here,

r is the radius of cylinder.

Write the equation of plane as follows.

x+2y+3z=1

Consider x and y as parameters and parameterize the plane as follows.

x=x,y=y,z=1x2y3x=x,y=y,z=1313x23y

Calculation of zx :

Take partial derivative for equation (1) with respect to x.

zx=x(1313x23y)=0130=13

Calculation of zy :

Take partial derivative for equation (1) with respect to y

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