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

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Chapter 7.9, Problem 9E
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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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Comparing Different Orders of Integration In Exercises 7-12, set up the integrals for both orders of integration and use the more convenient order to find the volume of the solid region bounded by the surface f(x, y) and the planes. See Example 2.

f ( x , y ) = y x 2 + y 2

Planes: z = 0 , y = x , y = 2 x , x = 2

To determine

To calculate: The volume of solid region bounded by surface f(x,y)=yx2+y2 and plane: z=0,y=x,y=2x,x=2.

Explanation

Given information:

The provided equation of surface f(x,y)=yx2+y2 and plane is z=0,y=x,y=2x,x=2.

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 surface,

f(x,y)=yx2+y2

The region R:

z=0,y=x,y=2x,x=2

The following table represent x and y coordinate of function y=2x,

x-coordinate y-coordinate y=2x (x,y)-coordinate
0 0 (0,0)
2 4 (2,4)

The following table represent x and y coordinate of function y=x,

x-coordinate y-coordinate y=x (x,y)-coordinate
0 0 (0,0)
2 2 (2,2)

The graph of region R z=0,y=x,y=2x,x=2

For order of limit, the bounds for x are 0≤x≤2 and bounds for y are x≤y≤2x.

The volume for the solid region is,

Volume=∫02∫x2x(yx2+y2)dydx

For order of limit the bounds for y are 0≤y≤4 and bounds for x are y2≤x≤y.

The volume for the solid region is,

Volume=∫04∫y2y(yx2+y2)dxdy

Integration is independent from order of limit

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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
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Single Variable Calculus

Ford and Torok (2008) found that motivational signs were effective in increasing physical activity on a college...

Statistics for The Behavioral Sciences (MindTap Course List)

General: Roll Two Dice You roll two fair dice, a green one and a red one. (a) Are the outcomes on the dice inde...

Understanding Basic Statistics

Expand each expression in Exercises 122. (y3+2y2+y)(y2+2y1)

Finite Mathematics

In Exercises 55 and 56, use the inclusion-exclusion formula to answer each application. 56. A college finds tha...

Mathematical Excursions (MindTap Course List)

For Problems 79-99, answer the question with an algebraic expression. Objective 3 The length of a line segment ...

Intermediate Algebra

In problems 37-44, perform the indicated operations and simplify. 42.

Mathematical Applications for the Management, Life, and Social Sciences

A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying dat...

Probability and Statistics for Engineering and the Sciences

Use the binomial series to expand the function as a power series. State the radius of convergence. 33. 1(2+x)3

Single Variable Calculus: Early Transcendentals

Draw 45 and – 45 in standard position and then show that sin (–45) = –sin 45.

Trigonometry (MindTap Course List)

From the following information, determine the number of days of each loan. Loan Dale Due Date Number of Days 33...

Contemporary Mathematics for Business & Consumers

In Exercises 21 to 24, state whether the lines are parallel, perpendicular, the same coincident, or none of the...

Elementary Geometry For College Students, 7e

Prove that the equalities in Exercises hold for all in . Assume only the basic postulates for and those prop...

Elements Of Modern Algebra

Write each number in decimal form: 746103

Elementary Technical Mathematics

Finding Surface Area Using Technology In Exercises 21-26, write a double integral that represents the surface a...

Calculus: Early Transcendental Functions (MindTap Course List)

Finding a Limit of a Trigonometric Function In Exercises 27-36, find the limit of the trigonometric function. l...

Calculus of a Single Variable

PERSONNEL SELECTION Jacobs & Johnson, an accounting firm, employs 14 accountants, of whom 8 are CPAs. If a dele...

Finite Mathematics for the Managerial, Life, and Social Sciences

Suppose that r=12 cm and h=15 cm in the right circular cylinder. Find the exact and approximate a lateral area....

Elementary Geometry for College Students

The area of a region D enclosed by a curve C, such as that pictured in question 1, may be evaluated using the l...

Study Guide for Stewart's Multivariable Calculus, 8th

Orbital SpeedIn Exercises 6568, use the result of Exercise 64 to find the speed necessary for the given circula...

Multivariable Calculus

Converting to Polar Coordinates In Exercises 2932, use polar coordinates to set up and evaluate the double inte...

Calculus

Graphical Reasoning Consider the function f(x)=ax(xb)2. Determine the effect on the graph of f as a and b are c...

Calculus (MindTap Course List)

True or False: y = xex is a solution to y=y+yx.

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

Explain the distinction between naturalistic observation and contrived observation, and describe the situations...

Research Methods for the Behavioral Sciences (MindTap Course List)

A partial computer output from a regression analysis follows. a. Compute the missing entries in this output. b....

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

Let x denote the duration of a randomly selected pregnancy (the time elapsed between conception and birth). Acc...

Introduction To Statistics And Data Analysis

What pattern of results is needed to provide convincing evidence that behavior changed when the phase was chang...

Research Methods for the Behavioral Sciences (MindTap Course List)

Americans spend nearly 7 billion on Halloween costumes and decorations (The Wall Street Journal, October 27, 20...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Graph the function defined by the equation fx=ex-e-x2 from x=-2 to x=2. The function is called the hyperbolic s...

College Algebra (MindTap Course List)

Solve each of the following equations using the addition principle of equality. Check each answer. C5716=5716

Mathematics For Machine Technology

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