(a) Explanation Given: The solid region is given as: Formula used: The volume of the solid in cylindrical coordinates is calculated using the formula: V = ∫ θ 1 θ 2 ∫ r 1 r 2 ∫ z 1 z 2 r d z d r d θ The rectangular conversion equations for the cylindrical coordinates are given as: x = r cos θ y = r sin θ z = z Calculation: The limits of r , θ and z for the bounded solid is as follows: The limits for z are calculated as: 2 ≤ z ≤ 8 − x 2 − y 2 2 ≤ z ≤ 8 − r 2 The limits for r are calculated as: 0 ≤ r ≤ 2 The limits for θ are calculated as: 0 ≤ θ ≤ 2 π Put the limits of r , θ and z in the formula for the volume, V = ∫ 0 2 π ∫ 0 2 ∫ 2 8 − r 2 r d z d r d θ On solving the above integral, the volume of the bounded solid is calculated to be: V = ∫ 0 2 π ∫ 0 2 ∫ 2 8 − r 2 r d z d r d θ = ∫ 0 2 π ∫ 0 2 ( ∫ 2 8 − r 2 r d z ) d r d θ = ∫ 0 2 π ∫ 0 2 ( r [ z ] 2 8 − r 2 ) d r d θ = ∫ 0 2 π ∫ 0 2 r ( 8 − r 2 − 2 ) d r d θ Simplifying further, V = ∫ 0 2 π ∫ 0 2 r 8 − r 2 d r d θ − ∫ 0 2 π ∫ 0 2 2 r d r d θ = ∫ 0 2 π ( (b) To determine To calculate: The volume of the solid bounded below by the plane z = 2 and above the sphere x 2 + y 2 + z 2 = 8 using spherical coordinates.

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ISBN: 9781337275347

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14, Problem 18PS

(a)

To determine

To calculate: The volume of the solid bound below by the plane z=2 and above the sphere x2+y2+z2=8 using cylindrical coordinates.

(b)

To determine

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using spherical coordinates.

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Chapter 14, Problem 16PS

Chapter 15.1, Problem 1E

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The volume of the solid bound below by the plane z = 2 and above the sphere x 2 + y 2 + z 2 = 8 using cylindrical coordinates.

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To calculate: The volume of the solid bound below by the plane z=2 and above the sphere x2+y2+z2=8 using cylindrical coordinates.

To calculate: The volume of the solid bounded below by the plane z=2 and above the sphere x2+y2+z2=8 using spherical coordinates.

Want to see the full answer?

Chapter 14 Solutions