(a) To derive the formula for the volume of a sphere of radius R in a new way. Show that the volume of a thin spherical shell of inner radius r and thickness Δ r is approximately 4 π r 2 Δ r

Question

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Answer 1

Question

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

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Answer 2

Question

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

Expert Solution & Answer

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Answer 3

Question

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

Expert Solution & Answer

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Answer 4

Question

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

Expert Solution & Answer

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Answer 5

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

Answer 6

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

Answer 7

Chapter 6.4, Problem 62E

To determine

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

Answer 8

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

Answer 9

To determine

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

Answer 10

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

Answer 11

To determine

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

Answer 12

Expert Solution & Answer

Answer 13

Expert Solution & Answer

Answer 14

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Answer 15

Ch. 6.1 - Prob. 1PQ Ch. 6.1 - Prob. 2PQ Ch. 6.1 - Prob. 3PQ Ch. 6.1 - Prob. 4PQ Ch. 6.1 - Prob. 5PQ Ch. 6.1 - Prob. 6PQ Ch. 6.1 - Prob. 1E Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E

(a) To derive the formula for the volume of a sphere of radius R in a new way. Show that the volume of a thin spherical shell of inner radius r and thickness Δ r is approximately 4 π r 2 Δ r

Solution Summary: The author illustrates how the volume of a thin spherical shell is given by the product of the surface area and the thickness.

Concept explainers

Videos

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

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Chapter 6 Solutions

(a) To derive the formula for the volume of a sphere of radius R in a new way. Show that the volume of a thin spherical shell of inner radius r and thickness Δ r is approximately 4 π r 2 Δ r

Solution Summary: The author illustrates how the volume of a thin spherical shell is given by the product of the surface area and the thickness.

Concept explainers

Videos

(a) To derive the formula for the volume of a sphere of radius R in a new way. Show that the volume of a thin spherical shell of inner radius r and thickness Δr is approximately 4πr2Δr

(b) To derive the formula for the volume of a sphere of radius R in a new way. Approximate the volume by decomposing the sphere of radius R into N thin spherical shells of thickness Δr=R/N

(c) To derive the formula for the volume of a sphere of radius R in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.

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Chapter 6 Solutions

(a)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the volume of a thin spherical shell of inner radius $r$ and thickness $Δ r$ is approximately $4 π r^{2} Δ r$

(b)

To derive the formula for the volume of a sphere of radius $R$ in a new way.

Approximate the volume by decomposing the sphere of radius $R$ into $N$ thin spherical shells of thickness $Δ r = R / N$

(c)

To derive the formula for the volume of a sphere of radius $R$ in a new way. Show that the approximation is a Riemann sum that converges to an integral and evaluate the integral.