   Chapter 12.6, Problem 44E

Chapter
Section
Textbook Problem

# Sketch the region bounded by the paraboloids z = x2 + y2 and z = 2 − x2 − y2.

To determine

To sketch: The region bounded by the paraboloids z=x2+y2 and z=2x2y2 .

Explanation

Given data:

Paraboloid equations are z=x2+y2 and z=2x2y2 .

Formula used:

Consider the standard equation of an elliptic paraboloid along the z axis.

zc=x2a2+y2b2 (1)

Consider the given surface equation.

z=x2+y2 (2)

Modify the equation as follows.

z1=x21+y21 (3)

By comparing equation (3) with (1), the given surface equation satisfies the equation of an elliptic paraboloid along the z axis.

Find a, b, and c by comparing equation (3) with equation (1).

a2=1a=1a=±1

The value of a is ±1 .

b2=1b=1b=±1

The value of b is ±1 .

c=1

The value of c is 1.

Case i:

Let x=k .

Substitute k for x in equation (1),

z=k2+y2

The trace of this expression represents a parabola in the positive z direction.

Case ii:

Let y=k

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