Thomas' Calculus (14th Edition)
14th Edition
ISBN: 9780134438986
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Chapter 15.5, Problem 19E
To determine
Find the value of the
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1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
Chapter 15 Solutions
Thomas' Calculus (14th Edition)
Ch. 15.1 - In Exercises 1-14. evaluate the iterated...Ch. 15.1 - Evaluating Iterated Integrals
In Exercises 1-14....Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14, evaluate the iterated...Ch. 15.1 - In Exercises 1-14. evaluate the iterated...
Ch. 15.1 - In Exercises 1-14. evaluate the iterated...Ch. 15.1 - In Exercises 1-14. evaluate the iterated...Ch. 15.1 - In Exercises 1–14, evaluate the iterated...Ch. 15.1 - In Exercises 1–14, evaluate the iterated...Ch. 15.1 - Find all values of the constant c so that
Ch. 15.1 - Find all values of the constant c so that
Ch. 15.1 - In Exercises 17-24, evaluate the double integral...Ch. 15.1 - In Exercises 17-24, evaluate the double integral...Ch. 15.1 - In Exercises 17-24, evaluate the double integral...Ch. 15.1 - In Exercises 17–24, evaluate the double integral...Ch. 15.1 - In Exercises 17–24, evaluate the double integral...Ch. 15.1 - In Exercises 17–24, evaluate the double integral...Ch. 15.1 - In Exercises 17–24, evaluate the double integral...Ch. 15.1 - In Exercises 17–24, evaluate the double integral...Ch. 15.1 - In Exercises 25 and 26, integrate f over the given...Ch. 15.1 - In Exercises 25 and 26, integrate f over the given...Ch. 15.1 - In Exercises 27 and 28, sketch the solid whose...Ch. 15.1 - In Exercises 27 and 28, sketch the solid whose...Ch. 15.1 - Find the volume of the region hounded above by the...Ch. 15.1 - Find the volume of the region bounded above by the...Ch. 15.1 - Find the volume of the region bounded above by the...Ch. 15.1 - Find the volume of the region bounded above by the...Ch. 15.1 - Find the volume of the region bounded above by the...Ch. 15.1 - Find the volume of the region bounded above by the...Ch. 15.1 - Find a value of the constant k so that
Ch. 15.1 - Evaluate .
Ch. 15.1 - Use Fubini’s Theorem to evaluate
.
Ch. 15.1 - Use Fubini’s Theorem to evaluate
Ch. 15.1 - Use a software application to compute the...Ch. 15.1 - Prob. 40ECh. 15.2 - In Exercises 1-8, sketch the described regions of...Ch. 15.2 - In Exercises 1-8, sketch the described regions of...Ch. 15.2 - In Exercises 1-8, sketch the described regions of...Ch. 15.2 - In Exercises 1-8, sketch the described regions of...Ch. 15.2 - In Exercises 1-8, sketch the described regions of...Ch. 15.2 - Prob. 6ECh. 15.2 - Prob. 7ECh. 15.2 - Prob. 8ECh. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - In Exercises 9-18, write an iterated integral for...Ch. 15.2 - In Exercises 9-18, write an iterated integral for...Ch. 15.2 - In Exercises 9–18, write an iterated integral for ...Ch. 15.2 - Finding Regions of Integration and Double...Ch. 15.2 - Finding Regions of Integration and Double...Ch. 15.2 - In Exercises 19–24, sketch the region of...Ch. 15.2 - In Exercises 19–24, sketch the region of...Ch. 15.2 - In Exercises 19–24, sketch the region of...Ch. 15.2 - Finding Regions of Integration and Double...Ch. 15.2 - In Exercises 25-28, integrate f over the given...Ch. 15.2 - In Exercises 25-28, integrate f over the given...Ch. 15.2 - In Exercises 25–28, integrate f over the given...Ch. 15.2 - Prob. 28ECh. 15.2 - Each of Exercises 29−32 gives an integral over a...Ch. 15.2 - Each of Exercises 29−32 gives an integral over a...Ch. 15.2 - Each of Exercises 29–32 gives an integral over a...Ch. 15.2 - Prob. 32ECh. 15.2 - In Exercises 33–46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - Prob. 36ECh. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - Prob. 40ECh. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - Prob. 45ECh. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 33-46, sketch the region of...Ch. 15.2 - In Exercises 47-56, sketch the region of...Ch. 15.2 - In Exercises 47-56, sketch the region of...Ch. 15.2 - In Exercises 47-56, sketch the region of...Ch. 15.2 - In Exercises 47-56, sketch the region of...Ch. 15.2 - In Exercises 47-56, sketch the region of...Ch. 15.2 - Prob. 53ECh. 15.2 - In Exercises 47-56, sketch the region of...Ch. 15.2 - In Exercises 47–56, sketch the region of...Ch. 15.2 - In Exercises 47–56, sketch the region of...Ch. 15.2 - Find the volume of the region bounded above by the...Ch. 15.2 - Find the volume of the solid that is bounded above...Ch. 15.2 - Find the volume of the solid whose base is the...Ch. 15.2 - Find the volume of the solid in the first octant...Ch. 15.2 - Find the volume of the solid in the first octant...Ch. 15.2 - Find the volume of the solid cut from the first...Ch. 15.2 - Find the volume of the wedge cut from the first...Ch. 15.2 - Find the volume of the solid cut from the square...Ch. 15.2 - Find the volume of the solid that is bounded on...Ch. 15.2 - Find the volume of the solid bounded on the front...Ch. 15.2 - In Exercises 67 and 68, sketch the region of...Ch. 15.2 - In Exercises 67 and 68, sketch the region of...Ch. 15.2 - Prob. 69ECh. 15.2 - Prob. 70ECh. 15.2 - Prob. 71ECh. 15.2 - Integrals over Unbounded Regions
Improper double...Ch. 15.2 - Prob. 73ECh. 15.2 - Prob. 74ECh. 15.2 - Prob. 75ECh. 15.2 - Prob. 76ECh. 15.2 - Noncircular cylinder A solid right (noncircular)...Ch. 15.2 - Prob. 78ECh. 15.2 - Prob. 79ECh. 15.2 - Minimizing a double integral What region R in the...Ch. 15.2 - Prob. 81ECh. 15.2 - Prob. 82ECh. 15.2 - Prob. 83ECh. 15.2 - Improper double integral Evaluate the improper...Ch. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - Prob. 2ECh. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - Prob. 4ECh. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - Prob. 7ECh. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - Prob. 10ECh. 15.3 - In Exercises 1-12, sketch the region bounded by...Ch. 15.3 - Prob. 12ECh. 15.3 - The integrals and sums of integrals in Exercises...Ch. 15.3 - The integrals and sums of integrals in Exercises...Ch. 15.3 - The integrals and sums of integrals in Exercises...Ch. 15.3 - The integrals and sums of integrals in Exercises...Ch. 15.3 - The integrals and sums of integrals in Exercises...Ch. 15.3 - The integrals and sums of integrals in Exercises...Ch. 15.3 - Find the average value of f(x, y) = sin(x + y)...Ch. 15.3 - Which do you think will be larger, the average...Ch. 15.3 - Find the average height of the paraboloid z = x2 +...Ch. 15.3 - Find the average value of f(x, y) = 1/(xy) over...Ch. 15.3 - Prob. 23ECh. 15.3 - Prob. 24ECh. 15.3 - Bacterium population If f(x, y) = (10,000ey)/ (1 +...Ch. 15.3 - Regional population If f(x, y) = 100 (y + 1)...Ch. 15.3 - Prob. 27ECh. 15.3 - Prob. 28ECh. 15.3 - Prob. 29ECh. 15.3 - Prob. 30ECh. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 - In Exercises 1-8, describe the given region in...Ch. 15.4 -
In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - Prob. 16ECh. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9-22, change the Cartesian integral...Ch. 15.4 - In Exercises 9–22, change the Cartesian integral...Ch. 15.4 - In Exercises 9–22, change the Cartesian integral...Ch. 15.4 - In Exercises 23-26, sketch the region of...Ch. 15.4 - In Exercises 23–26, sketch the region of...Ch. 15.4 - In Exercises 23–26, sketch the region of...Ch. 15.4 - In Exercises 23–26, sketch the region of...Ch. 15.4 - Find the area of the region cut from the first...Ch. 15.4 - Cardioid overlapping a circle Find the area of the...Ch. 15.4 - One leaf of a rose Find the area enclosed by one...Ch. 15.4 - Prob. 30ECh. 15.4 - Prob. 31ECh. 15.4 - Overlapping cardioids Find the area of the region...Ch. 15.4 - In polar coordinates, the average value of a...Ch. 15.4 - In polar coordinates, the average value of a...Ch. 15.4 - Prob. 35ECh. 15.4 - In polar coordinates, the average value of a...Ch. 15.4 - Converting to a polar integral Integrate over the...Ch. 15.4 - Converting to a polar integral Integrate over the...Ch. 15.4 - Volume of noncircular right cylinder The region...Ch. 15.4 - Prob. 40ECh. 15.4 - Converting to polar integrals
The usual way to...Ch. 15.4 - Converting to a polar integral Evaluate the...Ch. 15.4 - Existence Integrate the function f(x, y) = 1/(1 −...Ch. 15.4 - Area formula in polar coordinates Use the double...Ch. 15.4 - Prob. 45ECh. 15.4 - Area Suppose that the area of a region in the...Ch. 15.4 - Evaluate the integral , where R is the region...Ch. 15.4 - Evaluate the integral where R is the region...Ch. 15.5 - Evaluate the integral in Example 3, taking F(x, y,...Ch. 15.5 - Volume of rectangular solid Write six different...Ch. 15.5 - Volume of tetrahedron Write six different iterated...Ch. 15.5 - Volume of solid Write six different iterated...Ch. 15.5 - Volume enclosed by paraboloids Let D be the region...Ch. 15.5 - Volume inside paraboloid beneath a plane Let D be...Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
7.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
8.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
9.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
10.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
11.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
12.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
13.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
14.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
15.
Ch. 15.5 - Prob. 16ECh. 15.5 - Evaluate the integrals in Exercises 7–20.
17.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
18.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
19.
Ch. 15.5 - Evaluate the integrals in Exercises 7–20.
20.
Ch. 15.5 - Here is the region of integration of the...Ch. 15.5 - Here is the region of integration of the...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Prob. 32ECh. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - The region cut from the solid elliptical cylinder...Ch. 15.5 - Find the volumes of the regions in Exercises...Ch. 15.5 - In Exercises 37–40, find the average value of F(x,...Ch. 15.5 - In Exercises 37–40, find the average value of F(x,...Ch. 15.5 - In Exercises 37–40, find the average value of F(x,...Ch. 15.5 - In Exercises 37–40, find the average value of F(x,...Ch. 15.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 15.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 15.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 15.5 - Evaluate the integrals in Exercises 41–44 by...Ch. 15.5 - Finding an upper limit of an iterated integral...Ch. 15.5 - Ellipsoid For what value of c is the volume of the...Ch. 15.5 - Minimizing a triple integral What domain D in...Ch. 15.5 - Maximizing a triple integral What domain D in...Ch. 15.6 - Finding a center of mass find the center of mass...Ch. 15.6 - Finding moments of inertia Find the moments of...Ch. 15.6 - Finding a centroid Find the centroid of the region...Ch. 15.6 - Finding a centroid Find the centroid of the...Ch. 15.6 - Finding a centroid Find the centroid of the region...Ch. 15.6 - Finding a centroid Find the centroid of the region...Ch. 15.6 - Finding moments of inertia Find the moment of...Ch. 15.6 - Prob. 8ECh. 15.6 - The centroid of an infinite region Find the...Ch. 15.6 - Prob. 10ECh. 15.6 - Finding a moment of inertia Find the moment of...Ch. 15.6 - Prob. 12ECh. 15.6 - Finding a center of mass Find the center of mass...Ch. 15.6 - Finding a center of mass and moment of inertia...Ch. 15.6 - Center of mass, moment of inertia Find the center...Ch. 15.6 - Prob. 16ECh. 15.6 - Center of mass, moment of inertia Find the center...Ch. 15.6 - Prob. 18ECh. 15.6 - Center of mass, moments of inertia Find the center...Ch. 15.6 - Prob. 20ECh. 15.6 - Moments of inertia Find the moments of inertia of...Ch. 15.6 - Moments of inertia The coordinate axes in the...Ch. 15.6 - Prob. 23ECh. 15.6 - Center of mass A solid of constant density is...Ch. 15.6 - a. Center of mass Find the center of mass of a...Ch. 15.6 - Prob. 26ECh. 15.6 - Moment of inertia about a line A wedge like the...Ch. 15.6 - Prob. 28ECh. 15.6 - In Exercises 29 and 30, find
the mass of the...Ch. 15.6 - In Exercises 29 and 30, find
a. the mass of the...Ch. 15.6 - Prob. 31ECh. 15.6 - In Exercises 31 and 32, find
the mass of the...Ch. 15.6 - Mass Find the mass of the solid bounded by the...Ch. 15.6 - Mass Find the mass of the solid region bounded by...Ch. 15.6 - The Parallel Axis Theorem Let Lc.m. be a line...Ch. 15.6 - Prob. 36ECh. 15.6 - Prob. 37ECh. 15.6 - Prob. 38ECh. 15.6 - Joint Probability Density Functions
For Exercises...Ch. 15.6 - Prob. 40ECh. 15.6 - Joint Probability Density Functions
For Exercises...Ch. 15.6 - Prob. 42ECh. 15.6 - Prob. 43ECh. 15.6 - The following formula defines a joint probability...Ch. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - Prob. 3ECh. 15.7 - Prob. 4ECh. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - Prob. 6ECh. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - Prob. 8ECh. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - In Exercises 1–12, sketch the region described by...Ch. 15.7 - In Exercises 13−22, sketch the region described by...Ch. 15.7 - Prob. 14ECh. 15.7 - In Exercises 13−22, sketch the region described by...Ch. 15.7 - Prob. 16ECh. 15.7 - In Exercises 13−22, sketch the region described by...Ch. 15.7 - Prob. 18ECh. 15.7 - Prob. 19ECh. 15.7 - In Exercises 13−22, sketch the region described by...Ch. 15.7 - In Exercises 13−22, sketch the region described by...Ch. 15.7 - In Exercises 13−22, sketch the region described by...Ch. 15.7 - Evaluate the cylindrical coordinate integrals in...Ch. 15.7 - Evaluate the cylindrical coordinate integrals in...Ch. 15.7 - Evaluate the cylindrical coordinate integrals in...Ch. 15.7 - Evaluate the cylindrical coordinate integrals in...Ch. 15.7 - Evaluate the cylindrical coordinate integrals in...Ch. 15.7 - Evaluate the cylindrical coordinate integrals in...Ch. 15.7 - The integrals we have seen so far suggest that...Ch. 15.7 - The integrals we have seen so far suggest that...Ch. 15.7 - The integrals we have seen so far suggest that...Ch. 15.7 - The integrals we have seen so far suggest that...Ch. 15.7 - Let D be the region bounded below by the plane z =...Ch. 15.7 - Let D be the region bounded below by the cone and...Ch. 15.7 - Give the limits of integration for evaluating the...Ch. 15.7 - Convert the integral
to an equivalent integral in...Ch. 15.7 - In Exercises 37–42, set up the iterated integral...Ch. 15.7 - In Exercises 37–42, set up the iterated integral...Ch. 15.7 - In Exercises 37–42, set up the iterated integral...Ch. 15.7 - In Exercises 37–42, set up the iterated integral...Ch. 15.7 - Prob. 41ECh. 15.7 - In Exercises 37–42, set up the iterated integral...Ch. 15.7 - Evaluate the spherical coordinate integrals in...Ch. 15.7 - Evaluate the spherical coordinate integrals in...Ch. 15.7 - Evaluate the spherical coordinate integrals in...Ch. 15.7 - Evaluate the spherical coordinate integrals in...Ch. 15.7 - Prob. 47ECh. 15.7 - Prob. 48ECh. 15.7 - Prob. 49ECh. 15.7 - The previous integrals suggest there are preferred...Ch. 15.7 - The previous integrals suggest there are preferred...Ch. 15.7 - The previous integrals suggest there are preferred...Ch. 15.7 - Let D be the region in Exercise 33. Set up the...Ch. 15.7 - Let D be the region bounded below by the cone and...Ch. 15.7 - In Exercises 55–60, (a) find the spherical...Ch. 15.7 - In Exercises 55–60, (a) find the spherical...Ch. 15.7 - In Exercises 55–60, (a) find the spherical...Ch. 15.7 - Prob. 58ECh. 15.7 - In Exercises 55–60, (a) find the spherical...Ch. 15.7 - In Exercises 55–60, (a) find the spherical...Ch. 15.7 - Prob. 61ECh. 15.7 - Let D be the region in the first octant that is...Ch. 15.7 - Let D be the smaller cap cut from a solid ball of...Ch. 15.7 - Let D be the solid hemisphere x2 + y2 + z2 ≤ 1, z ...Ch. 15.7 - Find the volumes of the solids in Exercises...Ch. 15.7 - Prob. 66ECh. 15.7 - Prob. 67ECh. 15.7 - Find the volumes of the solids in Exercises...Ch. 15.7 - Find the volumes of the solids in Exercises...Ch. 15.7 - Find the volumes of the solids in Exercises...Ch. 15.7 - Sphere and cones Find the volume of the portion of...Ch. 15.7 - Prob. 72ECh. 15.7 - Prob. 73ECh. 15.7 - Cone and planes Find the volume of the solid...Ch. 15.7 - Cylinder and paraboloid Find the volume of the...Ch. 15.7 - Cylinder and paraboloids Find the volume of the...Ch. 15.7 - Cylinder and cones Find the volume of the solid...Ch. 15.7 - Sphere and cylinder Find the volume of the region...Ch. 15.7 - Prob. 79ECh. 15.7 - Cylinder and planes Find the volume of the region...Ch. 15.7 - Region trapped by paraboloids Find the volume of...Ch. 15.7 - Prob. 82ECh. 15.7 - Prob. 83ECh. 15.7 - Sphere and paraboloid Find the volume of the...Ch. 15.7 - Prob. 85ECh. 15.7 - Prob. 86ECh. 15.7 - Prob. 87ECh. 15.7 - Find the average value of the function f(ρ, ϕ, θ)...Ch. 15.7 - Prob. 89ECh. 15.7 - Prob. 90ECh. 15.7 - Prob. 91ECh. 15.7 - Prob. 92ECh. 15.7 - Prob. 93ECh. 15.7 - Centroid Find the centroid of the region cut from...Ch. 15.7 - Prob. 95ECh. 15.7 - Prob. 96ECh. 15.7 - Prob. 97ECh. 15.7 - Prob. 98ECh. 15.7 - Prob. 99ECh. 15.7 - Prob. 100ECh. 15.7 - Prob. 101ECh. 15.7 - Prob. 102ECh. 15.7 - Density of center of a planet A planet is in the...Ch. 15.7 - Prob. 104ECh. 15.7 - Prob. 105ECh. 15.7 - Prob. 106ECh. 15.7 - Prob. 107ECh. 15.7 - Prob. 108ECh. 15.8 - Solve the system
for x and y in terms of u and v....Ch. 15.8 - Prob. 2ECh. 15.8 - Solve the system
for x and y in terms of u and v....Ch. 15.8 - Solve the system
for x and y in terms of u and v....Ch. 15.8 - Prob. 5ECh. 15.8 - Use the transformation in Exercise 1 to evaluate...Ch. 15.8 - Use the transformation in Exercise 3 to evaluate...Ch. 15.8 - Prob. 8ECh. 15.8 - Let R be the region in the first quadrant of the...Ch. 15.8 - Find the Jacobian of the transformation and...Ch. 15.8 - Polar moment of inertia of an elliptical plate A...Ch. 15.8 - Prob. 12ECh. 15.8 - Use the transformation in Exercise 2 to evaluate...Ch. 15.8 - Use the transformation x = u + (1/2)v, y = v to...Ch. 15.8 - Use the transformation x = u/v, y = uv to evaluate...Ch. 15.8 - Prob. 16ECh. 15.8 - Prob. 17ECh. 15.8 - Volume of an ellipsoid Find the volume of the...Ch. 15.8 - Evaluate
over the solid ellipsoid D,
(Hint: Let...Ch. 15.8 - Let D be the region in xyz-space defined by the...Ch. 15.8 - Find the Jacobian ∂(x, y)/∂(u, v) of the...Ch. 15.8 - Find the Jacobian of the transformation
Ch. 15.8 - Prob. 23ECh. 15.8 - Prob. 24ECh. 15.8 - Prob. 25ECh. 15.8 - Prob. 26ECh. 15.8 - Inverse transform The equations x = g(u, v), y =...Ch. 15.8 - Prob. 28ECh. 15 - Prob. 1GYRCh. 15 - How are double integrals evaluated as iterated...Ch. 15 - Prob. 3GYRCh. 15 - How can you change a double integral in...Ch. 15 - Prob. 5GYRCh. 15 - Prob. 6GYRCh. 15 - How are double and triple integrals in rectangular...Ch. 15 - Prob. 8GYRCh. 15 - How are triple integrals in cylindrical and...Ch. 15 - Prob. 10GYRCh. 15 - Prob. 11GYRCh. 15 - Prob. 1PECh. 15 - Prob. 2PECh. 15 - In Exercises 1–4, sketch the region of integration...Ch. 15 - Prob. 4PECh. 15 - Prob. 5PECh. 15 - Prob. 6PECh. 15 - In Exercises 5–8, sketch the region of integration...Ch. 15 - Prob. 8PECh. 15 - Prob. 9PECh. 15 - Evaluate the integrals in Exercises 9–12.
10.
Ch. 15 - Prob. 11PECh. 15 - Prob. 12PECh. 15 - Prob. 13PECh. 15 - Area bounded by lines and parabola Find the area...Ch. 15 - Prob. 15PECh. 15 - Prob. 16PECh. 15 - Prob. 17PECh. 15 - Prob. 18PECh. 15 - Evaluate the integrals in Exercises 19 and 20 by...Ch. 15 - Prob. 20PECh. 15 - Integrating over a lemniscate Integrate the...Ch. 15 - Prob. 22PECh. 15 - Prob. 23PECh. 15 - Prob. 24PECh. 15 - Evaluate the integrals in Exercises 23–26.
25.
Ch. 15 - Prob. 26PECh. 15 - Prob. 27PECh. 15 - Volume Find the volume of the solid that is...Ch. 15 - Prob. 29PECh. 15 - Average value Find the average value of ρ over the...Ch. 15 - Cylindrical to rectangular coordinates Convert
to...Ch. 15 - Rectangular to cylindrical coordinates (a) Convert...Ch. 15 - Prob. 33PECh. 15 - Prob. 34PECh. 15 - Cylindrical to rectangular coordinates Set up an...Ch. 15 - Prob. 36PECh. 15 - Spherical versus cylindrical coordinates Triple...Ch. 15 - Prob. 38PECh. 15 - Prob. 39PECh. 15 - Prob. 40PECh. 15 - Prob. 41PECh. 15 - Prob. 42PECh. 15 - Polar moment Find the polar moment of inertia...Ch. 15 - Prob. 44PECh. 15 - Prob. 45PECh. 15 - Prob. 46PECh. 15 - Prob. 47PECh. 15 - Prob. 48PECh. 15 - Centroid Find the centroid of the region in the...Ch. 15 - Prob. 50PECh. 15 - Prob. 51PECh. 15 - Centroid Find the centroid of the plane region...Ch. 15 - Prob. 53PECh. 15 - Prob. 54PECh. 15 - Prob. 1AAECh. 15 - Prob. 2AAECh. 15 - Prob. 3AAECh. 15 - Prob. 4AAECh. 15 - Prob. 5AAECh. 15 - Prob. 6AAECh. 15 - Prob. 7AAECh. 15 - Prob. 8AAECh. 15 - Two paraboloids Find the volume of the region...Ch. 15 - Prob. 10AAECh. 15 - Prob. 11AAECh. 15 - Prob. 12AAECh. 15 - Prob. 13AAECh. 15 - Prob. 14AAECh. 15 - Minimizing polar inertia A thin plate of constant...Ch. 15 - Prob. 16AAECh. 15 - Mass and polar inertia of a counterweight The...Ch. 15 - Prob. 18AAECh. 15 - Prob. 19AAECh. 15 - Prob. 20AAECh. 15 - Prob. 21AAECh. 15 - Prob. 22AAECh. 15 - Prob. 23AAECh. 15 - Prob. 24AAECh. 15 - A parabolic rain gauge A bowl is in the shape of...Ch. 15 - Water in a satellite dish A parabolic satellite...Ch. 15 - Prob. 27AAECh. 15 - Prob. 28AAE
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- write it down for better understanding pleasearrow_forward1. Suppose F(t) gives the temperature in degrees Fahrenheit t minutes after 1pm. With a complete sentence, interpret the equation F(10) 68. (Remember this means explaining the meaning of the equation without using any mathy vocabulary!) Include units. (3 points) =arrow_forward2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below. a. Evaluate f(-3). If you have multiple steps, be sure to connect your expressions with EQUALS SIGNS. (3 points)arrow_forward
- 4c Consider the function f(x) = 10x + 4x5 - 4x³- 1. Enter the general antiderivative of f(x)arrow_forwardA tank contains 60 kg of salt and 2000 L of water. Pure water enters a tank at the rate 8 L/min. The solution is mixed and drains from the tank at the rate 11 L/min. Let y be the number of kg of salt in the tank after t minutes. The differential equation for this situation would be: dy dt y(0) =arrow_forwardSolve the initial value problem: y= 0.05y + 5 y(0) = 100 y(t) =arrow_forward
- y=f'(x) 1 8 The function f is defined on the closed interval [0,8]. The graph of its derivative f' is shown above. How many relative minima are there for f(x)? O 2 6 4 00arrow_forward60! 5!.7!.15!.33!arrow_forward• • Let > be a potential for the vector field F = (−2 y³, −6 xy² − 4 z³, −12 yz² + 4 2). Then the value of sin((-1.63, 2.06, 0.57) – (0,0,0)) is - 0.336 -0.931 -0.587 0.440 0.902 0.607 -0.609 0.146arrow_forward
- The value of cos(4M) where M is the magnitude of the vector field with potential ƒ = e² sin(лy) cos(π²) at x = 1, y = 1/4, z = 1/3 is 0.602 -0.323 0.712 -0.816 0.781 0.102 0.075 0.013arrow_forwardThere is exactly number a and one number b such that the vector field F = conservative. For those values of a and b, the value of cos(a) + sin(b) is (3ay + z, 3ayz + 3x, −by² + x) is -0.961 -0.772 -1.645 0.057 -0.961 1.764 -0.457 0.201arrow_forwardA: Tan Latitude / Tan P A = Tan 04° 30'/ Tan 77° 50.3' A= 0.016960 803 S CA named opposite to latitude, except when hour angle between 090° and 270°) B: Tan Declination | Sin P B Tan 052° 42.1'/ Sin 77° 50.3' B = 1.34 2905601 SCB is alway named same as declination) C = A + B = 1.35 9866404 S CC correction, A+/- B: if A and B have same name - add, If different name- subtract) = Tan Azimuth 1/Ccx cos Latitude) Tan Azimuth = 0.737640253 Azimuth = S 36.4° E CAzimuth takes combined name of C correction and Hour Angle - If LHA is between 0° and 180°, it is named "west", if LHA is between 180° and 360° it is named "east" True Azimuth= 143.6° Compass Azimuth = 145.0° Compass Error = 1.4° West Variation 4.0 East Deviation: 5.4 Westarrow_forward
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