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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.9, Problem 6E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding the Volume of a Solid Region In Exercises 1-6, find the volume of the solid region bounded in the first octant by the plane. See Example 1.

z = 1 − 4 x − 2 y

To determine

To calculate: The volume of solid region bounded in first octane and the plane z=1−4x−2y.

Explanation

Given information:

The provided equation of plane is z=1−4x−2y.

Formula used:

The procedure to calculate volume of surface z=f(x,y),

Step-1: Write the equation of surface in the form z=f(x,y)

Step-2: Sketch the projected region R in the x-y-plane.

Step-3: Determine the order of limits of integration.

Step-4: Evaluate the volume of solid region,

Volume=∫mn∫abf(x,y)dxdy

Here, the projected region is R:

m≤y≤n and a≤x≤b

Calculation:

Consider equation of plane,

z=1−4x−2y

The following table shown different coordinate of (x,y,z) for z=1−4x−2y.

x-Coordinate y-Coordinate z-Coordinate (x,y,z) Coordinate
0 0 1 (0,0,2)
0 12 0 (0,12,0)
14 0 0 (14,0,0)

The graph of function z=1−4x−2y in first octant shown below.

Substitute, 0 for z in equation of plane z=1−4x−2y.

1−4x−2y=04x+2y=1y=12−2x

The following table shown different coordinate of (x,y) for y=12−2x.

x-Coordinate y-Coordinate (x,y) Coordinate
14 0 (14,0)
0 12 (0,12)

The graph of equation y=12−2x is

One way to set up the double integration is to choose x as the outer variable with that choice. The bounds for x are 0≤x≤14 and bounds for y are 0≤y≤12−2x and another choose to set up the double integration is to choose y as the outer variable. with that choice, The bounds for y are 0≤y≤12 and bounds for x are 0≤x≤y2−14

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 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Single Variable Calculus

In problems 23-58, perform the indicated operations and simplify. 40.

Mathematical Applications for the Management, Life, and Social Sciences

For Problems 5-54, perform the following operations with real numbers. Objectives 3-6 115+345

Intermediate Algebra

The Pew Forum on Religion and Public Life reported on Dec. 9, 2009, that in a survey of 2003 American adults, 2...

Probability and Statistics for Engineering and the Sciences

Solve the equations in Exercises 126. 2x3x2x3x+1=0

Finite Mathematics

Find all solutions in radians using exact values only. sin2xcos3x+cos2xsin3x=1

Trigonometry (MindTap Course List)

22. Magda Leon is applying for a home mortgage with a monthly PITI of $724. She currently has a gross income of...

Contemporary Mathematics for Business & Consumers

Given a figure, which of the following pairs of transformations leads to an image that repeats the original fig...

Elementary Geometry For College Students, 7e

Draw the graph of the line through the given point with the given slope: (0,4),m=0

Elementary Technical Mathematics

a. Prove part a. of Theorem 1.34. b. Prove part b. of Theorem 1.34. Theorem 1.34 Distributive Properties Let A...

Elements Of Modern Algebra

Find a number that provides a counterexample to show that the given statement is false. For all numbers x, xx.

Mathematical Excursions (MindTap Course List)

Finding an Indefinite Integral In Exercises 53-62, find the indefinite integral. x(13x2)4dx

Calculus: Early Transcendental Functions (MindTap Course List)

Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal p...

Single Variable Calculus: Early Transcendentals

Electrical Resistance The resistance R of a certain type of resistor is R=0.001T44T+100 where R is measured in ...

Calculus of a Single Variable

In Exercises 47 to 50, use the fact that the area of the polygon is unique. In the right triangle, find the len...

Elementary Geometry for College Students

In Exercises 1926, pivot the system about the circled element. [013241562|434]

Finite Mathematics for the Managerial, Life, and Social Sciences

Precalculus or Calculus In Exercises 3-6.decide whether the problem can be solved using precalculus or whether ...

Calculus

The equation of the plane tangent to x2 + 2y2 + 2z2 = 5 at (−1, −1, 1) is: −(x + 1) − (y + 1) + (z − 1) = 0 −(x...

Study Guide for Stewart's Multivariable Calculus, 8th

Projectile Motion In Exercises 41 and 42, use the model for projectile motion, assuming there is no air resista...

Multivariable Calculus

True or False: If f(x) is continuous and decreasing, f(n) = an for all n = 1, 2, 3, …, and

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

In addition to the key words, you should also be able to define each of the following terms: Descriptive resear...

Research Methods for the Behavioral Sciences (MindTap Course List)

On a syndicated television show the two hosts often create the impression that they strongly disagree about whi...

Statistics for Business & Economics, Revised (MindTap Course List)

Given the workpiece or tool diameters and the revolutions per minute, determine the cutting speeds in the follo...

Mathematics For Machine Technology

Rationalize each numerator. 2+h-2h

College Algebra (MindTap Course List)

The grade point average for college students is based on a weighted mean computation. For most colleges, the gr...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Each year J.D. Power and Associates surveys new car owners 90 days after they purchase their cars. This data is...

Introduction To Statistics And Data Analysis

Why are studies that examine the effects of aging not considered true experiments?

Research Methods for the Behavioral Sciences (MindTap Course List)

Related Rates In your own words, state the guidelines for solving related-rate problems.

Calculus (MindTap Course List)

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