Explanation Prove that ∬ S x 2 d S = ∬ S y 2 d S = ∬ S z 2 d S using symmetry of the sphere with respect to the x z , x y and the y z planes and then show that ∬ S x 2 d S = = 4 3 π R 4 by adding the integrals Given : Let S be the sphere of radius R centered at the origin. Formula used : x 2 + y 2 + z 2 = R 2 Proof : Since the sphere is symmetric with respect to x z and y z planes, we can interchange x and y for the integral for ∬ S x 2 d S does not change the value of the integral and hence the result is ∬ S y 2 d S . Similarly, ∬ S y 2 d S is equal to ∬ S z 2 d S Hence ∬ S x 2 d S = ∬ S y 2 d S

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 17.4, Problem 28E

To determine

To prove:

Explain using symmetry:

$\iint_{S} x^{2} d S = \iint_{S} y^{2} d S = \iint_{S} z^{2} d S$ and then show that $\iint_{S} x^{2} d S = = \frac{4}{3} π R^{4}$ by adding the integrals

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Chapter 17.4, Problem 27E

Chapter 17.4, Problem 29E

Chapter 17 Solutions

CALCULUS (CLOTH)

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

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To prove : Explain using symmetry: ∬ S x 2 d S = ∬ S y 2 d S = ∬ S z 2 d S and then show that ∬ S x 2 d S = = 4 3 π R 4 by adding the integrals

Solution Summary: The author explains how to prove symmetry of the sphere with respect to x, y and z planes by adding the integrals.

Videos

To prove:

Explain using symmetry:

$\iint_{S} x^{2} d S = \iint_{S} y^{2} d S = \iint_{S} z^{2} d S$ and then show that $\iint_{S} x^{2} d S = = \frac{4}{3} π R^{4}$ by adding the integrals

Want to see the full answer?

Chapter 17 Solutions

To prove : Explain using symmetry: ∬ S x 2 d S = ∬ S y 2 d S = ∬ S z 2 d S and then show that ∬ S x 2 d S = = 4 3 π R 4 by adding the integrals

Solution Summary: The author explains how to prove symmetry of the sphere with respect to x, y and z planes by adding the integrals.

Videos

To prove: Explain using symmetry: ∬Sx2dS=∬Sy2dS=∬Sz2dS and then show that ∬Sx2dS==43πR4 by adding the integrals

Want to see the full answer?

Chapter 17 Solutions

To prove:

Explain using symmetry:

$\iint_{S} x^{2} d S = \iint_{S} y^{2} d S = \iint_{S} z^{2} d S$ and then show that $\iint_{S} x^{2} d S = = \frac{4}{3} π R^{4}$ by adding the integrals