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Volume In Exercises 21–24, use a triple integral to find thevolume of the solid bounded by the graphs of the equations. z = 4 − x 2 , y = 4 − x 2 first octant Volume In Exercises 21–24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = 4 − x 2 , y = 4 − x 2 first octant

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Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770

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Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770
Chapter 14.6, Problem 21E
Textbook Problem
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Volume In Exercises 21–24, use a triple integral to find thevolume of the solid bounded by the graphs of the equations.

z = 4 x 2 ,   y = 4 x 2 first octant

Volume In Exercises 21–24, use a triple integral to find the volume of the solid bounded by the graphs of the equations.

z = 4 x 2 ,   y = 4 x 2 first octant

To determine

To calculate: The volume of the solid bounded by the graphs of the equations

z=4x2, y=4x2, first octant using a triple integral.

Explanation of Solution

Given: The solid is bounded by the graphs of the equations z=4x2, y=4x2, first octant.

Formula used:

dz=z+C

dy=y+C

xn dx=xn+1n+1+C

Calculation:

In Figure, it can be observed that the solid is bounded below by the xy- plane and above by the paraboloid.

For the paraboloid z=4x2, the bounds for z will be 0z4x2.

For the paraboloid y=4x2, the bounds for y will be 0y4x2.

Choosing dy dx as the order of integration, the bounds of y and x will be 0y4x2 and

0x2.

Therefore, the volume of the solid is given by

V=QdV=0204x204x2dz dy dx

For the first integration, keeping x and y as constants and integrating with respect to z we get,

V=0204x2(04x2dz) dy dx=0204x2(z)04x2

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Chapter 14 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)
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The center of mass of the...Ch. 14.4 - Proof Prove the following Theorem of Pappus: Let R...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 114, find the...Ch. 14.5 - Finding Surface Areathe area of the surface given...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 114, find the...Ch. 14.5 - Finding Surface Area In Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding SurfaceArea In Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 17-20, find the...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using Technology In Exercises...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - WRITING ABOUT CONCEPTS Surface Area State the...Ch. 14.5 - WRITING ABOUT CONCEPTS Surface Area Answer each...Ch. 14.5 - WRITING ABOUT CONCEPTS Surface Area Will the...Ch. 14.5 - HOW DO YOU SEE IT? Consider the surface...Ch. 14.5 - Product DesignA company produces a spherical...Ch. 14.5 - Modeling Data A company builds a warehouse with...Ch. 14.5 - Surface Area Find the surface area of the solid of...Ch. 14.5 - Surface Area Show that the surface area of the...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral Using...Ch. 14.6 - Evaluating a Triple Iterated Integral Using...Ch. 14.6 - Setting Up a Triple Integral In Exercises 1116,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - VolumeIn Exercises 1720, use a triple integral to...Ch. 14.6 - Volume In Exercises 1720, use a triple integral to...Ch. 14.6 - Volume In Exercises 2124, use a triple integral to...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 2124, use a triple integral to...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 35 and 36, the...Ch. 14.6 - Orders of Integration In Exercises 35 and 36, the...Ch. 14.6 - Center of Mass In Exercises 37-40, Find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, Find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, Find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, Find the mass...Ch. 14.6 - Center of MassIn Exercises 41 and 42, set up the...Ch. 14.6 - Center of MassIn Exercises 41 and 42, set up the...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About ItThe center of mass of a solid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find Ix,...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of InertiaIn Exercises 53- 56, find Ix,Iy,...Ch. 14.6 - Moments of InertiaIn Exercises 53- 56, find Ix,Iy,...Ch. 14.6 - Moments of InertiaIn Exercises 57 and 58, verify...Ch. 14.6 - Moments of Inertia In Exercises 57 and 58, verify...Ch. 14.6 - Moments of InertiaIn Exercises 59 and 60, set up a...Ch. 14.6 - Moments of InertiaIn Exercises 59 and 60, set up a...Ch. 14.6 - Setting Up Triple IntegralsIn Exercises 61 and 62,...Ch. 14.6 - Setting Up Triple IntegralsIn Exercises 61 and 62,...Ch. 14.6 - Average ValueIn Exercises 63-66, find the average...Ch. 14.6 - Average ValueIn Exercises 63-66, find the average...Ch. 14.6 - Average ValueIn Exercises 63-66, find the average...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - WRITING ABOUT CONCEPTS Triple Integral Define a...Ch. 14.6 - Moment of Inertia Determine whether the momentof...Ch. 14.6 - EXPLORING CONCEPTS (continued) Think About It...Ch. 14.6 - HOW DO YOU SEE IT? Consider two solids of equal...Ch. 14.6 - Maximizing a Triple Integral Find the solid region...Ch. 14.6 - Finding a Value Solve for a in the triple integral...Ch. 14.6 - PUTNAM EXAM CHALLENGE Evaluate...Ch. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14.7 - Evaluating an Iterated IntegralIn Exercises 16,...Ch. 14.7 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14.7 - Evaluating an Iterated IntegralIn Exercises 16,...Ch. 14.7 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14.7 - Evaluating a Triple Iterated Integral Using...Ch. 14.7 - Evaluating a Triple Iterated Integral Using...Ch. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - VolumeIn Exercises 912, sketch the solid region...Ch. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - Converting Coordinates In Exercises 41-44, convert...Ch. 14.7 - Converting Coordinates In Exercises 1316, convert...Ch. 14.7 - Converting Coordinates In Exercises 41-44, convert...Ch. 14.7 - Volume In Exercises 1722, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 1722, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Mass In Exercises 21 and 22, use cylindrical...Ch. 14.7 - Mass In Exercises 21 and 22, use cylindrical...Ch. 14.7 - Using Cylindrical CoordinatesIn Exercises 23-28,...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Using Cylindrical CoordinatesIn Exercises 23-28,...Ch. 14.7 - Moment of Inertia In Exercises 29 and 30, use...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Mass In Exercises 35 and 36, use spherical...Ch. 14.7 - Mass In Exercises 35 and 36, use spherical...Ch. 14.7 - Center of Mass In Exercises 37 and 38, use...Ch. 14.7 - Center of MassIn Exercises 37 and 38, use...Ch. 14.7 - Moment of Inertia In Exercises 39 and 40, use...Ch. 14.7 - Moment of Inertia In Exercises 39 and 40, use...Ch. 14.7 - Coordinate ConversionGive the equations for...Ch. 14.7 - Coordinate ConversionGive the equations for...Ch. 14.7 - Cylindrical FormGive the iterated form of the...Ch. 14.7 - Spherical FormGive the iterated form of the triple...Ch. 14.7 - EXPLORING CONCEPTS Using Coordinates Describe the...Ch. 14.7 - HOW DO YOU SEE IT? The solid is bounded below by...Ch. 14.7 - PUTNAM EXAM CHALLENGE Find the volume of the...Ch. 14.8 - CONCEPT CHECK Change of Variable When is it...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 18, rind the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 18, rind the...Ch. 14.8 - Finding a JacobianIn Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Using a TransformationIn Exercises 11-14, sketch...Ch. 14.8 - Verifying a Change of Variables In Exercises 15...Ch. 14.8 - Verifying a Change of Variables In Exercises 15...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - EXPLORING CONCEPTS Using a Transformation The...Ch. 14.8 - HOW DO YOU SEE IT? The region R is transformed...Ch. 14.8 - Using an Ellipse Consider the region R in the...Ch. 14.8 - Volume Use the result of Exercise 33 to find the...Ch. 14.8 - JacobianState the definition of the Jacobian.Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - PUTNAM EXAM CHALLENGE Let A be the area of the...Ch. 14 - Evaluating an Integral In Exercises 1 and 2,...Ch. 14 - Evaluating an Integral In Exercises 1 and 2,...Ch. 14 - Evaluating an Iterated Integral In Exercises 3-6,...Ch. 14 - Evaluating an Iterated Integral In Exercises 3-6,...Ch. 14 - Evaluating an Iterated Integral In Exercises 36,...Ch. 14 - Evaluating an Iterated Integral In Exercises 3-6,...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...Ch. 14 - Switching the Order of Integration In Exercises...Ch. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Switching the Order of Integration In Exercises...Ch. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Evaluating a Double Integral In Exercises 15 and...Ch. 14 - Evaluating a Double Integral In Exercises 15 and...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Average Value In Exercises 21 and 22, find the...Ch. 14 - Average Value In Exercises 21 and 22, find the...Ch. 14 - Average Temperature The temperature in degrees...Ch. 14 - Average Profit A firms profit P (in dollars) from...Ch. 14 - Converting to Polar Coordinates In Exercises 25...Ch. 14 - Converting to Polar Coordinates In Exercises 25...Ch. 14 - Volume In Exercises 27 and 28, use a double...Ch. 14 - Volume In Exercises 27 and 28, use a double...Ch. 14 - Area In Exercises 29 and 30, use a double integral...Ch. 14 - Area In Exercises 29 and 30, use a double integral...Ch. 14 - Area In Exercises 31 and 32, sketch a graph of the...Ch. 14 - Area In Exercises 31 and 32, sketch a graph of the...Ch. 14 - Area and Volume Consider the region R in the...Ch. 14 - Converting to Polar Coordinates Write the sum of...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding Moments of Inertia and Radii of Gyration...Ch. 14 - Finding Moments of Inertia and Radii of Gyration...Ch. 14 - Finding Surface Area In Exercises 43-46, find the...Ch. 14 - Finding Surface Area In Exercises 43-46, find the...Ch. 14 - Finding Surface Area In Exercises 43-46, find the...Ch. 14 - Finding Surface Area of the surface given by In...Ch. 14 - Building Design A new auditorium is built with a...Ch. 14 - Surface Area The roof over the stage of an open...Ch. 14 - Evaluating a Triple Iterated Integral In...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Volume In Exercises 55 and 56, use a triple...Ch. 14 - Volume In Exercises 55 and 56, use a triple...Ch. 14 - Changing the Order of Integration In Exercises 57...Ch. 14 - Center of Mass In Exercises 59 and 60, find the...Ch. 14 - Center of Mass In Exercises 59 and 60, find the...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating an Iterated Integral In Exercises 5962,...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Volume In Exercises 67 and 68, use cylindrical...Ch. 14 - VolumeIn Exercises 69 and 70, use spherical...Ch. 14 - Finding a Jacobian In Exercises 6770, find the...Ch. 14 - Finding a Jacobian In Exercises 71-74, find the...Ch. 14 - Finding a Jacobian In Exercises 71-74, find the...Ch. 14 - Finding a Jacobian In Exercises 71-74, find the...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Volume Find the volume of the solid of...Ch. 14 - Surface Area Let a, b, c, and d be positive real...Ch. 14 - Using a Change of Variables The figure shows the...Ch. 14 - Proof Prove that limn0101xnyndxdy=0.Ch. 14 - Deriving a Sum Derive Eulers famous result that...Ch. 14 - Evaluating a Double Integral Evaluate the integral...Ch. 14 - Evaluating Double Integrals Evaluate the integrals...Ch. 14 - Volume Show that the volume of a spherical block...Ch. 14 - Evaluating an Integral In Exercises 9 and 10,...Ch. 14 - Evaluating an Integral In Exercises 9 and 10,...Ch. 14 - Joint Density Function Consider the function...Ch. 14 - Volume Find the volume of the solid generated by...Ch. 14 - Proof The angle between a plane P and the xy-plane...Ch. 14 - Surface Area Use the result of Exercise 14 to...Ch. 14 - Sprinkler Consider a circular lawn with a radius...Ch. 14 - Volume The figure shows a solid bounded below by...

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