Explanation Given: A provided spherical solid is: Formula used: ∫ d z = z + C ∫ a 2 − y 2 d y = y 2 a 2 − y 2 + a 2 2 arcsin ( y a ) + C ; Integration table arcsin ( 1 ) = π 2 ∫ x n d x = x n + 1 n + 1 + C Calculation: As the sphere is symmetric about the origin, we determine the volume of the sphere in the first octant i.e., x ≥ 0 , y ≥ 0 , and z ≥ 0 . For the sphere z = a 2 − x 2 − y 2 , the bounds for z will be 0 ≤ z ≤ a 2 − x 2 − y 2 . The projection of the sphere onto the xy- plane is the quarter circle x 2 + y 2 = a 2 . Choosing d y d x as the order of integration, the bounds of y and x will be 0 ≤ y ≤ a 2 − x 2 and 0 ≤ x ≤ a . Therefore, the volume of the solid is given by V = 8 ∭ Q d V = 8 ∫ 0 a ∫ 0 a 2 − x 2 ∫ 0 a 2 − x 2 − y 2 d z d y d x For the first integration, keeping x and y as constant and integrating with respect to z we get, V = 8 ∫ 0 a ∫ 0 a 2 − x 2 ( ∫ 0 a 2 − x 2 − y 2 d z ) d y d x = 8 ∫ 0 a ∫ 0 a 2 − x 2 ( z ) 0 a</

Calculus: Early Transcendental Functions (MindTap Course List)

6th Edition

ISBN: 9781285774770

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14.6, Problem 19E

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To calculate: The volume of the solid using a triple integral.

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Chapter 14.6, Problem 18E

Chapter 14.6, Problem 20E

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Chapter 14 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)

Ch. 14.1 - Evaluate the iterated integral: 0433cosrdrd Ch. 14.1 - Prob. 1E Ch. 14.1 - Evaluate the integral: xx2yxdy Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Evaluate the integral: 04x2x2ydy Ch. 14.1 - Evaluate the integral: x3x(x2+3y2)dy Ch. 14.1 - Evaluate the integral: eyyylnxxdx;y0 Ch. 14.1 - Evaluate the integral: 1y21y2(x2+y2)dx Ch. 14.1 - Evaluate the integral: 0x2yeyxdy

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The volume of the solid using a triple integral.

Solution Summary: The author calculates the volume of the given solid using a triple integral.

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To calculate: The volume of the solid using a triple integral.

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Chapter 14 Solutions