Chapter 14: Q. 6 (page 1131)

Give three examples of quantities that may be computed by $\iint_{R} G (x, y) d A .$

Short Answer

Expert verified

Ans: The three examples of quantities that may be computed by $\iint_{R} G (x, y) d A .$ are:

1) Surface area is computed by the double integral.

2) The mass of lamina with density function $ρ (x, y, z)$ is computed by double integral.

3) The volume under the surface $z = z (x, y)$ is computed by the double integral.

Step by step solution

Step 1. Given information.

The objective is to provide three examples of quantities that may be computed by $\iint_{R} G (x, y) d A .$

Step 2. First example

The surface area is computed by the following double integral:

$\iint_{R} \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1} d A$ .

In this case,

$G (x, y) = \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1} .$

Step 3. The second example

The mass of lamina with density function $ρ (x, y, z)$ is computed by following double integral:

$\iint_{R} ρ (x, y, z (x, y)) \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1} d A$

In this case,

localid="1651053939013" $G (x, y) = ρ (x, y, z (x, y)) \sqrt{{(\frac{\partial z}{\partial x})}^{2} + {(\frac{\partial z}{\partial y})}^{2} + 1} .$

Step 4. The third example

The volume under the surface $z = z (x, y)$ is computed by following double integral:

\iint_{R} z (x, y) d A

In this case

$G (x, y) = z (x, y)$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Give three examples of quantities that may be computed by $\iint_{R} G (x, y) d A .$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First example

Step 3. The second example

Step 4. The third example

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Geometry

Probability and Statistics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Give three examples of quantities that may be computed by∬R G(x,y)dA.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. First example

Step 3. The second example

Step 4. The third example

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Geometry

Probability and Statistics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Give three examples of quantities that may be computed by $\iint_{R} G (x, y) d A .$