To prove: That G ( θ , ϕ ) = ( a cos θ sin ϕ , b sin θ sin ϕ , c cos ϕ ) is a parametrization of the given ellipsoid. To calculate: The volume of ellipsoid ( x a ) 2 + ( y b ) 2 + ( z c ) 2 = 1 by calculating surface integral of F = 1 3 〈 x , y , z 〉

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Answer 1

Question

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

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Find the surface area of the ellipsoid obtained by rotating the ellipse (=) = 1 about the x-axis.

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(b) A hole is drilled through the center of a sphere x² + y² + z² = R,², leaving a ring-shape solid. If the drill used is cylindrical in shape with radius R4, use polar coordinates to formulate the volume V of the remaining solid.

Answer 2

Question

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

Expert Solution & Answer

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Check out a sample textbook solution

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Answer 3

Question

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

Expert Solution & Answer

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Answer 4

Question

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

Expert Solution & Answer

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Check out a sample textbook solution

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Answer 5

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

Answer 6

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

Answer 7

Chapter 18, Problem 41CRE

To determine

To prove:

That $G (θ, ϕ) = (a \cos θ \sin ϕ, b \sin θ \sin ϕ, c \cos ϕ)$ is a parametrization of the given ellipsoid.

To calculate:

The volume of ellipsoid ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} + {(\frac{z}{c})}^{2} = 1$ by calculating surface integral of $F = \frac{1}{3} 〈 x, y, z 〉$

Answer 8

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Answer 9

Expert Solution & Answer

Answer 10

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Answer 11

Ch. 18.1 - Prob. 1PQ Ch. 18.1 - Prob. 2PQ Ch. 18.1 - Prob. 3PQ Ch. 18.1 - Prob. 4PQ Ch. 18.1 - Prob. 5PQ Ch. 18.1 - Prob. 1E Ch. 18.1 - Prob. 2E Ch. 18.1 - Prob. 3E Ch. 18.1 - Prob. 4E Ch. 18.1 - Prob. 5E

To prove: That G ( θ , ϕ ) = ( a cos θ sin ϕ , b sin θ sin ϕ , c cos ϕ ) is a parametrization of the given ellipsoid. To calculate: The volume of ellipsoid ( x a ) 2 + ( y b ) 2 + ( z c ) 2 = 1 by calculating surface integral of F = 1 3 〈 x , y , z 〉

Solution Summary: The author proves that G is a parametrization of the given ellipsoid by calculating surface integral of F = 1 3 x, y and z.

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Chapter 18 Solutions

To prove: That G ( θ , ϕ ) = ( a cos θ sin ϕ , b sin θ sin ϕ , c cos ϕ ) is a parametrization of the given ellipsoid. To calculate: The volume of ellipsoid ( x a ) 2 + ( y b ) 2 + ( z c ) 2 = 1 by calculating surface integral of F = 1 3 〈 x , y , z 〉

Solution Summary: The author proves that G is a parametrization of the given ellipsoid by calculating surface integral of F = 1 3 x, y and z.

Videos

To prove: That G(θ,ϕ)=(acosθsinϕ,bsinθsinϕ,ccosϕ) is a parametrization of the given ellipsoid. To calculate: The volume of ellipsoid (xa)2+(yb)2+(zc)2=1 by calculating surface integral of F=13〈x,y,z〉

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Chapter 18 Solutions