Explanation Given information: The lower portion cut from the sphere x 2 + y 2 + z 2 = 2 by the cone z = x 2 + y 2 Definition: The area of the smooth surface r ( u , v ) = f ( u , v ) i + g ( u , v ) j + h ( u , v ) k , a ≤ u ≤ b , c ≤ v ≤ d is A = ∬ R | r u × r v | d A = ∫ c d ∫ a b | r u × r v | d u d v . Calculation: Write the equations. z = x 2 + y 2 (1) x 2 + y 2 + z 2 = 2 (2) Take x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ and z = ρ cos ϕ . Substitute ρ sin ϕ cos θ for x , ρ sin ϕ sin θ for y and ρ cos ϕ for z in Equation (2). x 2 + y 2 + z 2 = ( ρ sin ϕ cos θ ) 2 + ( ρ sin ϕ sin θ ) 2 + ( ρ cos ϕ ) 2 = ρ 2 sin 2 ϕ cos 2 θ + ρ 2 sin 2 ϕ sin 2 θ + ρ 2 cos 2 ϕ = ρ 2 sin 2 ϕ ( cos 2 θ + sin 2 θ ) + ρ 2 cos 2 ϕ = ρ 2 ( sin 2 ϕ + cos 2 ϕ ) x 2 + y 2 + z 2 = ρ 2 ⇒ ρ = x 2 + y 2 + z 2 From Equation (2), ρ = x 2 + y 2 + z 2 = 2 . Substitute Equation (1) in Equation (2). x 2 + y 2 + z 2 = 2 z 2 + z 2 = 2 2 z 2 = 2 z 2 = 1 Since z ≥ 0 ⇒ ϕ = π 4 . For the lower portion of the sphere cut by the cone, ϕ = π . Then the parametrization of the surface is r ( ϕ , θ ) = ( 2 sin ϕ cos θ ) i + ( 2 sin ϕ sin θ ) j + ( 2 cos ϕ ) k , π 4 ≤ ϕ ≤ π , 0 ≤ θ ≤ 2 π (3) Differentiate Equation (3) with respect to ϕ . r ϕ ( ϕ , θ ) = ( 2 ( cos ϕ ) cos θ ) i + ( 2 ( cos ϕ ) sin θ ) j + ( 2 ( − sin ϕ ) ) k = ( 2 cos ϕ cos θ ) i + ( 2 cos ϕ sin θ ) j − ( 2 sin ϕ ) k Differentiate Equation (3) with respect to θ . r θ ( ϕ , θ ) = ( 2 sin ϕ ( − sin θ ) ) i + ( 2 sin ϕ cos θ ) j + ( 0 ) k = ( − 2 sin ϕ sin θ ) i + ( 2 sin ϕ cos θ ) j Find r ϕ × r θ

Thomas' Calculus: Early Transcendentals (14th Edition)

14th Edition

ISBN: 9780134439020

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 16.5, Problem 25E

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Chapter 16 Solutions

Thomas' Calculus: Early Transcendentals (14th Edition)

Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Evaluate ∫C (x + y) ds, where C is the...Ch. 16.1 - Prob. 10E

Ch. 16.1 - Evaluate ∫C (xy + y + z) ds along the curve r(t) =...Ch. 16.1 - Evaluate along the curve r(t) = (4 cos t)i + (4...Ch. 16.1 - Find the line integral of f(x, y, z) = x + y + z...Ch. 16.1 - Prob. 14E Ch. 16.1 - Prob. 15E Ch. 16.1 - Integrate over the path C1 followed by C2...Ch. 16.1 - Prob. 17E Ch. 16.1 - Prob. 18E Ch. 16.1 - Evaluate ∫C x ds, where C is the straight-line...Ch. 16.1 - Evaluate , where C is the straight-line segment x...Ch. 16.1 - Prob. 21E Ch. 16.1 - Find the line integral of f(x, y) = x − y + 3...Ch. 16.1 - Prob. 23E Ch. 16.1 - Prob. 24E Ch. 16.1 - Prob. 25E Ch. 16.1 - Evaluate , where C is given in the accompanying...Ch. 16.1 - Prob. 27E Ch. 16.1 - Prob. 28E Ch. 16.1 - Prob. 29E Ch. 16.1 - Prob. 30E Ch. 16.1 - Find the area of one side of the “winding wall”...Ch. 16.1 - Prob. 32E Ch. 16.1 - Prob. 33E Ch. 16.1 - Center of mass of a curved wire A wire of density ...Ch. 16.1 - Prob. 35E Ch. 16.1 - Prob. 36E Ch. 16.1 - Prob. 37E Ch. 16.1 - Prob. 38E Ch. 16.1 - Prob. 39E Ch. 16.1 - Prob. 40E Ch. 16.1 - Prob. 41E Ch. 16.1 - Prob. 42E Ch. 16.2 - Find the gradient fields of the functions in...Ch. 16.2 - Prob. 2E Ch. 16.2 - Prob. 3E Ch. 16.2 - Prob. 4E Ch. 16.2 - Prob. 5E Ch. 16.2 - Prob. 6E Ch. 16.2 - In Exercises 7−12, find the line integrals of F...Ch. 16.2 - Prob. 8E Ch. 16.2 - Prob. 9E Ch. 16.2 - Prob. 10E Ch. 16.2 - Line Integrals of Vector Fields In Exercises 7−12,...Ch. 16.2 - Prob. 12E Ch. 16.2 - Prob. 13E Ch. 16.2 - Prob. 14E Ch. 16.2 - In Exercises 13–16, find the line integrals along...Ch. 16.2 - Prob. 16E Ch. 16.2 - Prob. 17E Ch. 16.2 - Prob. 18E Ch. 16.2 - In Exercises 19–22, find the work done by F over...Ch. 16.2 - Prob. 20E Ch. 16.2 - Prob. 21E Ch. 16.2 - Prob. 22E Ch. 16.2 - Prob. 23E Ch. 16.2 - Prob. 24E Ch. 16.2 - Prob. 25E Ch. 16.2 - Prob. 26E Ch. 16.2 - Prob. 27E Ch. 16.2 - Prob. 28E Ch. 16.2 - Prob. 29E Ch. 16.2 - Prob. 30E Ch. 16.2 - Prob. 31E Ch. 16.2 - Prob. 32E Ch. 16.2 - In Exercises 31–34, find the circulation and flux...Ch. 16.2 - Prob. 34E Ch. 16.2 - Prob. 35E Ch. 16.2 - Prob. 36E Ch. 16.2 - Prob. 37E Ch. 16.2 - Prob. 38E Ch. 16.2 - Prob. 39E Ch. 16.2 - Find the circulation of the field F = yi + (x +...Ch. 16.2 - Prob. 41E Ch. 16.2 - Prob. 42E Ch. 16.2 - Prob. 43E Ch. 16.2 - Prob. 44E Ch. 16.2 - Prob. 45E Ch. 16.2 - Prob. 46E Ch. 16.2 - Prob. 47E Ch. 16.2 - Prob. 48E Ch. 16.2 - A field of tangent vectors Find a field G = P(x,...Ch. 16.2 - Prob. 50E Ch. 16.2 - Prob. 51E Ch. 16.2 - Prob. 52E Ch. 16.2 - Prob. 53E Ch. 16.2 - Work done by a radial force with constant...Ch. 16.2 - Prob. 55E Ch. 16.2 - Prob. 56E Ch. 16.2 - Prob. 57E Ch. 16.2 - Prob. 58E Ch. 16.2 - Circulation Find the circulation of F = 2xi + 2zj...Ch. 16.2 - Prob. 60E Ch. 16.2 - Prob. 61E Ch. 16.2 - Prob. 62E Ch. 16.3 - Which fields in Exercises 1–6 are conservative,...Ch. 16.3 - Prob. 2E Ch. 16.3 - Prob. 3E Ch. 16.3 - Prob. 4E Ch. 16.3 - Prob. 5E Ch. 16.3 - Prob. 6E Ch. 16.3 - Finding Potential Functions In Exercises 7–12,...Ch. 16.3 - In Exercises 7–12, find a potential function f...Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - Prob. 10E Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - Prob. 12E Ch. 16.3 - Prob. 13E Ch. 16.3 - Prob. 14E Ch. 16.3 - Prob. 15E Ch. 16.3 - Prob. 16E Ch. 16.3 - Prob. 17E Ch. 16.3 - Prob. 18E Ch. 16.3 - Prob. 19E Ch. 16.3 - Prob. 20E Ch. 16.3 - Prob. 21E Ch. 16.3 - Prob. 22E Ch. 16.3 - Prob. 23E Ch. 16.3 - Prob. 24E Ch. 16.3 - Prob. 25E Ch. 16.3 - Prob. 26E Ch. 16.3 - Prob. 27E Ch. 16.3 - Prob. 28E Ch. 16.3 - Work along different paths Find the work done by F...Ch. 16.3 - Prob. 30E Ch. 16.3 - Prob. 31E Ch. 16.3 - Integral along different paths Evaluate the line...Ch. 16.3 - Prob. 33E Ch. 16.3 - Prob. 34E Ch. 16.3 - Prob. 35E Ch. 16.3 - Prob. 36E Ch. 16.3 - Prob. 37E Ch. 16.3 - Gravitational field Find a potential function for...Ch. 16.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 16.4 - Prob. 2E Ch. 16.4 - Prob. 3E Ch. 16.4 - Prob. 4E Ch. 16.4 - Prob. 5E Ch. 16.4 - Prob. 6E Ch. 16.4 - Prob. 7E Ch. 16.4 - In Exercises 7–10, verify the conclusion of...Ch. 16.4 - Prob. 9E Ch. 16.4 - Prob. 10E Ch. 16.4 - Prob. 11E Ch. 16.4 - Prob. 12E Ch. 16.4 - Prob. 13E Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - Prob. 15E Ch. 16.4 - Prob. 16E Ch. 16.4 - Prob. 17E Ch. 16.4 - Prob. 18E Ch. 16.4 - Prob. 19E Ch. 16.4 - Prob. 20E Ch. 16.4 - Prob. 21E Ch. 16.4 - Prob. 22E Ch. 16.4 - Prob. 23E Ch. 16.4 - Prob. 24E Ch. 16.4 - Prob. 25E Ch. 16.4 - Prob. 26E Ch. 16.4 - Prob. 27E Ch. 16.4 - Prob. 28E Ch. 16.4 - Prob. 29E Ch. 16.4 - Prob. 30E Ch. 16.4 - Prob. 31E Ch. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Prob. 35E Ch. 16.4 - Prob. 36E Ch. 16.4 - Prob. 37E Ch. 16.4 - Prob. 38E Ch. 16.4 - Prob. 39E Ch. 16.4 - Prob. 40E Ch. 16.4 - Prob. 41E Ch. 16.4 - Prob. 42E Ch. 16.4 - Prob. 43E Ch. 16.4 - Prob. 44E Ch. 16.4 - Prob. 45E Ch. 16.4 - Prob. 46E Ch. 16.4 - Prob. 47E Ch. 16.4 - Prob. 48E Ch. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - Prob. 2E Ch. 16.5 - Prob. 3E Ch. 16.5 - Prob. 4E Ch. 16.5 - Prob. 5E Ch. 16.5 - Prob. 6E Ch. 16.5 - Prob. 7E Ch. 16.5 - Prob. 8E Ch. 16.5 - Prob. 9E Ch. 16.5 - Prob. 10E Ch. 16.5 - Prob. 11E Ch. 16.5 - Prob. 12E Ch. 16.5 - Prob. 13E Ch. 16.5 - Prob. 14E Ch. 16.5 - Prob. 15E Ch. 16.5 - Prob. 16E Ch. 16.5 - Prob. 17E Ch. 16.5 - Prob. 18E Ch. 16.5 - Prob. 19E Ch. 16.5 - Prob. 20E Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - Prob. 23E Ch. 16.5 - Prob. 24E Ch. 16.5 - Prob. 25E Ch. 16.5 - Prob. 26E Ch. 16.5 - Prob. 27E Ch. 16.5 - Prob. 28E Ch. 16.5 - Prob. 29E Ch. 16.5 - Prob. 30E Ch. 16.5 - Prob. 31E Ch. 16.5 - Prob. 32E Ch. 16.5 - Prob. 33E Ch. 16.5 - Prob. 34E Ch. 16.5 - Prob. 35E Ch. 16.5 - Prob. 36E Ch. 16.5 - Prob. 37E Ch. 16.5 - Prob. 38E Ch. 16.5 - Prob. 39E Ch. 16.5 - Prob. 40E Ch. 16.5 - Prob. 41E Ch. 16.5 - Find the area of the cap cut from the sphere x2 +...Ch. 16.5 - Prob. 43E Ch. 16.5 - Prob. 44E Ch. 16.5 - Prob. 45E Ch. 16.5 - Prob. 46E Ch. 16.5 - Prob. 47E Ch. 16.5 - Prob. 48E Ch. 16.5 - Prob. 49E Ch. 16.5 - Prob. 50E Ch. 16.5 - Prob. 51E Ch. 16.5 - Find the area of the surfaces in Exercises...Ch. 16.5 - Prob. 53E Ch. 16.5 - Prob. 54E Ch. 16.5 - Prob. 55E Ch. 16.5 - Prob. 56E Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 5E Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 7E Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 9E Ch. 16.6 - Prob. 10E Ch. 16.6 - Prob. 11E Ch. 16.6 - Prob. 12E Ch. 16.6 - Prob. 13E Ch. 16.6 - Prob. 14E Ch. 16.6 - Prob. 15E Ch. 16.6 - Integrate G(x, y, z) = x over the surface given by...Ch. 16.6 - Prob. 17E Ch. 16.6 - Integrate G(x, y, z) = x – y – z over the portion...Ch. 16.6 - Prob. 19E Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 21E Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 25E Ch. 16.6 - Prob. 26E Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 28E Ch. 16.6 - Prob. 29E Ch. 16.6 - Prob. 30E Ch. 16.6 - Prob. 31E Ch. 16.6 - Prob. 32E Ch. 16.6 - Prob. 33E Ch. 16.6 - Prob. 34E Ch. 16.6 - Prob. 35E Ch. 16.6 - Prob. 36E Ch. 16.6 - Prob. 37E Ch. 16.6 - Prob. 38E Ch. 16.6 - Prob. 39E Ch. 16.6 - Prob. 40E Ch. 16.6 - Prob. 41E Ch. 16.6 - Prob. 42E Ch. 16.6 - Prob. 43E Ch. 16.6 - Prob. 44E Ch. 16.6 - Prob. 45E Ch. 16.6 - Prob. 46E Ch. 16.6 - Prob. 47E Ch. 16.6 - Prob. 48E Ch. 16.6 - Prob. 49E Ch. 16.6 - Prob. 50E Ch. 16.7 - Prob. 1E Ch. 16.7 - Prob. 2E Ch. 16.7 - Prob. 3E Ch. 16.7 - Prob. 4E Ch. 16.7 - Prob. 5E Ch. 16.7 - Prob. 6E Ch. 16.7 - Prob. 7E Ch. 16.7 - Prob. 8E Ch. 16.7 - Prob. 9E Ch. 16.7 - In Exercises 7–12, use the surface integral in...Ch. 16.7 - Prob. 11E Ch. 16.7 - Prob. 12E Ch. 16.7 - Prob. 13E Ch. 16.7 - Prob. 14E Ch. 16.7 - Prob. 15E Ch. 16.7 - Evaluate where S is the hemisphere x2 + y2 + z2 =...Ch. 16.7 - Prob. 17E Ch. 16.7 - Prob. 18E Ch. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - Prob. 20E Ch. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - Prob. 22E Ch. 16.7 - Prob. 23E Ch. 16.7 - Prob. 24E Ch. 16.7 - Prob. 25E Ch. 16.7 - Verify Stokes’ Theorem for the vector field F =...Ch. 16.7 - Prob. 27E Ch. 16.7 - Prob. 28E Ch. 16.7 - Prob. 29E Ch. 16.7 - Prob. 30E Ch. 16.7 - Prob. 31E Ch. 16.7 - Does Stokes’ Theorem say anything special about...Ch. 16.7 - Prob. 33E Ch. 16.7 - Prob. 34E Ch. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 2E Ch. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 4E Ch. 16.8 - Prob. 5E Ch. 16.8 - Prob. 6E Ch. 16.8 - Prob. 7E Ch. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 9E Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 11E Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 13E Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 15E Ch. 16.8 - Prob. 16E Ch. 16.8 - Prob. 17E Ch. 16.8 - Prob. 18E Ch. 16.8 - Prob. 19E Ch. 16.8 - Prob. 20E Ch. 16.8 - Prob. 21E Ch. 16.8 - Prob. 22E Ch. 16.8 - Prob. 23E Ch. 16.8 - Prob. 24E Ch. 16.8 - Prob. 25E Ch. 16.8 - Prob. 26E Ch. 16.8 - Calculate the net outward flux of the vector...Ch. 16.8 - Prob. 28E Ch. 16.8 - Prob. 29E Ch. 16.8 - Prob. 30E Ch. 16.8 - Prob. 31E Ch. 16.8 - Prob. 32E Ch. 16.8 - Prob. 33E Ch. 16.8 - Green’s second formula (Continuation of Exercise...Ch. 16.8 - Prob. 35E Ch. 16.8 - Prob. 36E Ch. 16 - Prob. 1GYR Ch. 16 - Prob. 2GYR Ch. 16 - Prob. 3GYR Ch. 16 - Prob. 4GYR Ch. 16 - Prob. 5GYR Ch. 16 - Prob. 6GYR Ch. 16 - Prob. 7GYR Ch. 16 - Prob. 8GYR Ch. 16 - Prob. 9GYR Ch. 16 - Prob. 10GYR Ch. 16 - Prob. 11GYR Ch. 16 - Prob. 12GYR Ch. 16 - Prob. 13GYR Ch. 16 - Prob. 14GYR Ch. 16 - Prob. 15GYR Ch. 16 - Prob. 16GYR Ch. 16 - Prob. 17GYR Ch. 16 - Prob. 18GYR Ch. 16 - Prob. 1PE Ch. 16 - Prob. 2PE Ch. 16 - Prob. 3PE Ch. 16 - Prob. 4PE Ch. 16 - Prob. 5PE Ch. 16 - Prob. 6PE Ch. 16 - Prob. 7PE Ch. 16 - Prob. 8PE Ch. 16 - Prob. 9PE Ch. 16 - Prob. 10PE Ch. 16 - Prob. 11PE Ch. 16 - Area of a parabolic cap Find the area of the cap...Ch. 16 - Prob. 13PE Ch. 16 - Prob. 14PE Ch. 16 - Prob. 15PE Ch. 16 - Prob. 16PE Ch. 16 - Prob. 17PE Ch. 16 - Prob. 18PE Ch. 16 - Prob. 19PE Ch. 16 - Prob. 20PE Ch. 16 - Prob. 21PE Ch. 16 - Prob. 22PE Ch. 16 - Prob. 23PE Ch. 16 - Prob. 24PE Ch. 16 - Prob. 25PE Ch. 16 - Prob. 26PE Ch. 16 - Prob. 27PE Ch. 16 - Prob. 28PE Ch. 16 - Prob. 29PE Ch. 16 - Prob. 30PE Ch. 16 - Prob. 31PE Ch. 16 - Prob. 32PE Ch. 16 - Prob. 33PE Ch. 16 - Find potential functions for the fields in...Ch. 16 - Prob. 35PE Ch. 16 - Prob. 36PE Ch. 16 - Prob. 37PE Ch. 16 - Prob. 38PE Ch. 16 - Prob. 39PE Ch. 16 - Prob. 40PE Ch. 16 - Prob. 41PE Ch. 16 - Prob. 42PE Ch. 16 - Prob. 43PE Ch. 16 - Prob. 44PE Ch. 16 - Prob. 45PE Ch. 16 - Prob. 46PE Ch. 16 - Prob. 47PE Ch. 16 - Prob. 48PE Ch. 16 - Prob. 49PE Ch. 16 - Prob. 50PE Ch. 16 - Prob. 51PE Ch. 16 - Prob. 52PE Ch. 16 - Prob. 53PE Ch. 16 - Prob. 54PE Ch. 16 - Prob. 55PE Ch. 16 - Prob. 56PE Ch. 16 - Prob. 57PE Ch. 16 - Prob. 58PE Ch. 16 - Prob. 59PE Ch. 16 - Prob. 60PE Ch. 16 - Prob. 1AAE Ch. 16 - Prob. 2AAE Ch. 16 - Prob. 3AAE Ch. 16 - Prob. 4AAE Ch. 16 - Prob. 5AAE Ch. 16 - Prob. 6AAE Ch. 16 - Prob. 7AAE Ch. 16 - Find the mass of a helicoids r(r, ) = (r cos )i +...Ch. 16 - Prob. 9AAE Ch. 16 - Prob. 10AAE Ch. 16 - Prob. 11AAE Ch. 16 - Prob. 12AAE Ch. 16 - Archimedes’ principle If an object such as a ball...Ch. 16 - Prob. 14AAE Ch. 16 - Prob. 15AAE Ch. 16 - Prob. 16AAE Ch. 16 - Prob. 17AAE Ch. 16 - Prob. 18AAE Ch. 16 - Prob. 19AAE Ch. 16 - Prob. 20AAE Ch. 16 - Prob. 21AAE

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