 15.5PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.1AAE: VolumesSand pile: double and triple integrals The base of a sand pi...
 15.1PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.1QGY: Define the double integral of a function of two variables over a bo...
 15.2AAE: VolumesWater in a hemispherical bowl A hemispherical bowl of radius...
 15.2PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.2QGY: How are double integrals evaluated as iterated integrals? Does the ...
 15.3AAE: VolumesSolid cylindrical region between two planes Find the volume ...
 15.3PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.3QGY: How are double integrals used to calculate areas and average values...
 15.4AAE: VolumesSphere and paraboloid Find the volume of the region bounded ...
 15.4PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.4QGY: How can you change a double integral in rectangular coordinates int...
 15.5AAE: VolumesTwo paraboloids Find the volume of the region bounded above ...
 15.5QGY: Define the triple integral of a function ƒ(x, y, z) over a bounded ...
 15.6AAE: VolumesSpherical coordinates Find the volume of the region enclosed...
 15.6PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.6QGY: How are triple integrals in rectangular coordinates evaluated? How ...
 15.7AAE: VolumesHole in sphere A circular cylindrical hole is bored through ...
 15.7PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.7QGY: How are double and triple integrals in rectangular coordinates used...
 15.8AAE: VolumesSphere and cylinder Find the volume of material cut from the...
 15.8PE: Evaluating Double Iterated IntegralsIn Exercise, sketch the region ...
 15.8QGY: How are triple integrals defined in cylindrical and spherical coord...
 15.9AAE: VolumesTwo paraboloids Find the volume of the region enclosed by th...
 15.9PE: Evaluating Double Iterated IntegralsEvaluate the integrals in Exercise
 15.9QGY: How are triple integrals in cylindrical and spherical coordinates e...
 15.10AAE: VolumesCylinder and surface z = xy Find the volume of the region in...
 15.10PE: Evaluating Double Iterated IntegralsEvaluate the integrals in Exercise
 15.10QGY: How are substitutions in double integrals pictured as transformatio...
 15.11AAE: Changing the Order of IntegrationEvaluate the integral to form a do...
 15.11PE: Evaluating Double Iterated IntegralsEvaluate the integrals in Exercise
 15.11QGY: How are substitutions in triple integrals pictured as transformatio...
 15.12AAE: Changing the Order of Integrationa. Polar coordinates Show, by chan...
 15.12PE: Evaluating Double Iterated IntegralsEvaluate the integrals in Exercise
 15.13AAE: Changing the Order of IntegrationReducing a double to a single inte...
 15.13PE: Areas and Volumes Using Double IntegralsArea between line and parab...
 15.14AAE: Changing the Order of IntegrationTransforming a double integral to ...
 15.14PE: Areas and Volumes Using Double IntegralsArea bounded by lines and p...
 15.15AAE: Masses and MomentsMinimizing polar inertia A thin plate of constant...
 15.15PE: Areas and Volumes Using Double IntegralsVolume of the region under ...
 15.16AAE: Masses and MomentsPolar inertia of triangular plate Find the polar ...
 15.16PE: Areas and Volumes Using Double IntegralsVolume of the region under ...
 15.17AAE: Masses and MomentsMass and polar inertia of a counterweight The cou...
 15.17PE: Average ValuesFind the average value of ƒ(x, y) = xy over the regio...
 15.18AAE: Masses and MomentsCentroid of boomerang Find the centroid of the bo...
 15.18PE: Average ValuesFind the average value of ƒ(x, y) = xy over the regio...
 15.19AAE: Theory and ExamplesEvaluate
 15.19PE: Polar CoordinatesEvaluate the integrals in Exercise by changing to ...
 15.20AAE: Theory and ExamplesShow that
 15.22PE: Polar CoordinatesIntegrate overa. Triangular region The triangle wi...
 15.21AAE: Theory and ExamplesSuppose that f(x, y) can be written as a product...
 15.21PE: Polar CoordinatesIntegrating over lemniscate Integrate the function...
 15.22AAE: Theory and ExamplesLet Duƒ denote the derivative of in the directio...
 15.23AAE: Theory and ExamplesThe value of The gamma function, extends the fac...
 15.23PE: Evaluating Triple Iterated IntegralsEvaluate the integrals in Exerc...
 15.24AAE: Theory and ExamplesTotal electrical charge over circular plate The ...
 15.24PE: Evaluating Triple Iterated IntegralsEvaluate the integrals in Exerc...
 15.25AAE: A parabolic rain gauge A bowl is in the shape of the graph of z = x...
 15.25PE: Evaluating Triple Iterated IntegralsEvaluate the integrals in Exerc...
 15.26AAE: Theory and ExamplesWater in a satellite dish A parabolic satellite ...
 15.26PE: Evaluating Triple Iterated IntegralsEvaluate the integrals in Exerc...
 15.27AAE: Theory and ExamplesAn infinite halfcylinder Let D be the interior ...
 15.27PE: Volumes and Average Values Using Triple IntegralsVolume Find the vo...
 15.28AAE: Theory and ExamplesHypervolume We have learned that is the length o...
 15.28PE: Volumes and Average Values Using Triple IntegralsVolume Find the vo...
 15.29PE: Volumes and Average Values Using Triple IntegralsAverage value Find...
 15.30PE: Volumes and Average Values Using Triple IntegralsAverage value Find...
 15.31PE: Cylindrical and Spherical CoordinatesCylindrical to rectangular coo...
 15.32PE: Cylindrical and Spherical CoordinatesRectangular to cylindrical coo...
 15.33PE: Cylindrical and Spherical CoordinatesRectangular to spherical coord...
 15.34PE: Cylindrical and Spherical CoordinatesRectangular, cylindrical, and ...
 15.35PE: Cylindrical and Spherical CoordinatesCylindrical to rectangular coo...
 15.36PE: Cylindrical and Spherical CoordinatesRectangular to cylindrical coo...
 15.37PE: Cylindrical and Spherical CoordinatesSpherical versus cylindrical c...
 15.38PE: Cylindrical and Spherical CoordinatesMasses and MomentsFinding in s...
 15.39PE: Moment of inertia of a “thick” sphere Find the moment of inertia of...
 15.40PE: Moment of inertia of an apple Find the moment of inertia about the ...
 15.41PE: Centroid Find the centroid of the “triangular” region bounded by th...
 15.42PE: Centroid Find the centroid of the region between the parabola x + y...
 15.43PE: Polar moment Find the polar moment of inertia about the origin of a...
 15.44PE: Polar moment Find the polar moment of inertia about the center of a...
 15.45PE: Inertial moment Find the moment of inertia about the xaxis ofa thi...
 15.46PE: Plate with variable density Find the center of mass and the moments...
 15.47PE: Plate with variable density Find the mass and first moments about t...
 15.48PE: Triangles with same inertial moment Find the moment of inertia abou...
 15.49PE: Centroid Find the centroid of the region in the polar coordinate pl...
 15.50PE: Centroid Find the centroid of the region in the first quadrant boun...
 15.51PE: a. Centroid Find the centroid of the region in the polar coordinate...
 15.52PE: a. Centroid Find the centroid of the plane region defined by the po...
 15.53PE: SubstitutionsShow that if u = x – y and v = y, then
 15.54PE: SubstitutionsWhat relationship must hold between the constants a, b...
Solutions for Chapter 15: Thomas' Calculus: Early Transcendentals 13th Edition
Full solutions for Thomas' Calculus: Early Transcendentals  13th Edition
ISBN: 9780321884077
Solutions for Chapter 15
Get Full SolutionsThis textbook survival guide was created for the textbook: Thomas' Calculus: Early Transcendentals , edition: 13. Chapter 15 includes 92 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 92 problems in chapter 15 have been answered, more than 71248 students have viewed full stepbystep solutions from this chapter. Thomas' Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321884077.

Constant of variation
See Power function.

Course
See Bearing.

Cubic
A degree 3 polynomial function

Endpoint of an interval
A real number that represents one “end” of an interval.

Focal length of a parabola
The directed distance from the vertex to the focus.

Higherdegree polynomial function
A polynomial function whose degree is ? 3

Horizontal asymptote
The line is a horizontal asymptote of the graph of a function ƒ if lim x: q ƒ(x) = or lim x: q ƒ(x) = b

Inverse of a matrix
The inverse of a square matrix A, if it exists, is a matrix B, such that AB = BA = I , where I is an identity matrix.

Inverse secant function
The function y = sec1 x

Line of travel
The path along which an object travels

Linear regression line
The line for which the sum of the squares of the residuals is the smallest possible

Local maximum
A value ƒ(c) is a local maximum of ƒ if there is an open interval I containing c such that ƒ(x) < ƒ(c) for all values of x in I

Lower bound of f
Any number b for which b < ƒ(x) for all x in the domain of ƒ

Normal curve
The graph of ƒ(x) = ex2/2

Open interval
An interval that does not include its endpoints.

Phase shift
See Sinusoid.

Product of functions
(ƒg)(x) = ƒ(x)g(x)

Quartic function
A degree 4 polynomial function.

Radius
The distance from a point on a circle (or a sphere) to the center of the circle (or the sphere).

ycoordinate
The directed distance from the xaxis xzplane to a point in a plane (space), or the second number in an ordered pair (triple), pp. 12, 629.