Chapter 12.6, Problem 44E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Sketch the region bounded by the paraboloids *z* = *x*^{2} + *y*^{2} and *z* = 2 − *x*^{2} − *y*^{2}.

To determine

**To sketch**: The region bounded by the paraboloids

Explanation

**Given data:**

Paraboloid equations are

**Formula used:**

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

Consider the given surface equation.

Modify the equation as follows.

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).

The value of *a* is

The value of *b* is

The value of *c* is 1.

**Case i:**

Let

Substitute *k* for *x* in equation (1),

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

**Case ii:**

Let

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