   Chapter 16.6, Problem 26E

Chapter
Section
Textbook Problem

Find a parametric representation for the surface.26. The part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 1

To determine

To find: The parametric representation for the part of the plane z=x+3 that lies inside the cylinder x2+y2=1 .

Explanation

Given data:

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

z=x+3 (1)

The required part of the plane lies inside the cylinder x2+y2=1 .

As the equation of the cylinder is x2+y2=1 , the radius r of the cylinder is 1 and the limit of radius of the radius is 0r1 .

Consider the parametric equations for the cylinder x2+y2=1 as follows.

x=rcosθ,y=rsinθ,0r1,0θ2π

Calculation of parametric equations of the part of the plane z=x+3 :

As the required part of the plane z=x+3 lies inside the cylinder x2+y2=1 , the x and y parameters of the plane are same as the cylinder parameters.

x=rcosθ,y=rsinθ,0r1,0θ2π

Substitute rcosθ for x in equation (1) to obtain the z parameter of the required plane.

z=rcosθ+3=3+rcosθ

From the analysis, the parametric equations of the part of the plane z=x+3 are written as follows

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