(a) Explanation Given: Theregion is bounded by the curve y = a x 2 + 1 and y = 0 with x = 0 and x = 1 . Formula used: The volume of the (washer method about x axis) is V = ∫ a b π ( f ( x ) 2 − g ( x ) 2 ) d x , where f ( x ) and g ( x ) is mentioned above. The volume of the (shell method about y axis) is V = ∫ c d 2 π ( f ( x ) − g ( x ) ) d x , where f ( x ) and g ( x ) is mentioned above. Calculation: Let f ( x ) = a x 2 + 1 and g ( x ) = 0 . Section 1: Calculate volume of the solid when the region is revolved about the x axis. The curves are f ( x ) = a x 2 + 1 and g ( x ) = 0 . Let l = f ( x ) 2 − g ( x ) 2 . To calculate the volume of the solid, first calculate the value of l with f ( x ) = a x 2 + 1 and g ( x ) = 0 as follows, l = ( a x 2 + 1 ) 2 − ( 0 ) 2 = a 2 x 4 + 1 + 2 a x 2 Substitute the value of l = a 2 x 4 + 1 + 2 a x 2 and take x = 0 as the lower limit and x = 1 as the upper limit in the mentioned washer formula above. V 1 = ∫ 0 1 π ( a 2 x 4 + 1 + 2 a x 2 ) d x = π a 2 ∫ 0 1 x 4 d x + 2 a π ∫ 0 1 x 2 d x + π ∫ 0 1 d x = π a 2 5 [ x 5 ] 0 1 + 2 a π 3 [ x 3 ] 0 1 + π [ x ] 0 1 = π a 2 (b) To determine To find: The values of a for which the volumes of the solids V 1 and V 2 are equal when the region R is revolving about the x and y- axes.

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Question

Chapter 6.4, Problem 69E

(a)

To determine

To find: The volume V1 and V2 of the solids when the region R is revolving about the x and y axes.

(b)

To determine

To find: The values of a for which the volumes of the solids V1 and V2 are equal when the region R is revolving about the x and y-axes.

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Chapter 6.4, Problem 68E

Chapter 6.4, Problem 70E

Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 6.1 - A police officer leaves his station on a...Ch. 6.1 - Describe a possible motion of an object along a...Ch. 6.1 - Is the position s(t) a number or a function? For...Ch. 6.1 - Without doing further calculations, what are the...Ch. 6.1 - Prob. 5QC Ch. 6.1 - Prob. 6QC Ch. 6.1 - Explain the meaning of position, displacement, and...Ch. 6.1 - Suppose the velocity of an object moving along a...Ch. 6.1 - Given the velocity function v of an object moving...Ch. 6.1 - Prob. 4E

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The volume V 1 and V 2 of the solids when the region R is revolving about the x and y axes.

Solution Summary: The author calculates the volume of the solids when the region R is revolving about the x and y axes.

To find: The volume V1 and V2 of the solids when the region R is revolving about the x and y axes.

To find: The values of a for which the volumes of the solids V1 and V2 are equal when the region R is revolving about the x and y-axes.

Want to see the full answer?

Chapter 6 Solutions