Chapter 16.6, Problem 44E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the area of the surface.

**44.** The part of the surface *z* = 4 − 2*x*^{2} +* y* that lies above the triangle with vertices (0, 0), (1, 0), and (1, 1)

To determine

**To find:** The area of the part of the surface

Explanation

**Given data:**

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

The required surface lies above the triangle with vertices

**Formula used:**

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

Write the equation of part of the surface as follows.

**Calculation of**
**:**

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

**Calculation of**
**:**

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

**Calculation of surface area of plane:**

Substitute

Simplify the expression as follows.

As the required part of the surface lies above the triangle with vertices

The region *D* is a triangle area with the *x* limit as 0 to 1 and the limit *y* is 0 to *x*. Therefore, the limit of *x* and *y* are written as follows.

Apply the limits and rewrite the expression in equation (3) as follows.

Simplify the expression as follows

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P-6RECh-16 P-7RECh-16 P-8RECh-16 P-9RECh-16 P-10RECh-16 P-11RECh-16 P-12RECh-16 P-13RECh-16 P-14RECh-16 P-15RECh-16 P-16RECh-16 P-17RECh-16 P-18RECh-16 P-19RECh-16 P-20RECh-16 P-21RECh-16 P-22RECh-16 P-23RECh-16 P-24RECh-16 P-25RECh-16 P-27RECh-16 P-28RECh-16 P-29RECh-16 P-30RECh-16 P-31RECh-16 P-32RECh-16 P-33RECh-16 P-34RECh-16 P-35RECh-16 P-36RECh-16 P-37RECh-16 P-38RECh-16 P-39RECh-16 P-40RECh-16 P-41RECh-16 P-1PCh-16 P-2PCh-16 P-3PCh-16 P-5PCh-16 P-6P

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