5. (a) Consider a unit half-sphere occupying the region (r? + y? + z2)1/2 <1 and z20. A 'dog-bowl' shape is created by removing the conical regionz> A(z² + y²)!/² , for some A > 0.(i) Calculate the volume of this dog-bowl shape as a function of A, leaving your answer in a form that does not involve any trigonometric functions.(i) For what value of A is the volume of the dog-bowl shape the same as the volume of the conical section that was removed?(b) Let A denote the semi-infinite strip described by r -1 -1 is a real number and n is a positive integer.(i) Use the transformation u = y - z to rewrite I as an integral over a semi-infinite rectangular domain. Hence show that I = 0 if n is an odd integer.(ii) Assuming n is an even integer, use another substitution to determine I(a, n) in terms of the beta function.(ii) Relate your answer to the gamma function, and thus evaluate I(3/2, n) for even integers n, simplifying your answer as much as possible.

Answered: Image /qna-images/answer/cad8d1d9-34d8-43e4-b021-4d6a908fa2c9.jpg

5. (a) Consider a unit half-sphere occupying the region (r? + y? +z2)1/2 <1 and z 20. A 'dog-bowl' shape is created by removing the conical region z> A(z² + y²)/2 , for some A > 0. (i) Calculate the volume of this dog-bowl shape as a function of A, leaving your answer in a form that does not involve any trigonometric functions. (ii) For what value of A is the volume of the dog-bowl shape the same as the volume of the conical section that was removed? (b) Let A denote the semi-infinite strip described by a -1 0. Consider the integral

5. (a) Consider a unit half-sphere occupying the region (r? + y? +z2)1/2 <1 and z 20. A 'dog-bowl' shape is created by removing the conical region z> A(z² + y²)/2 , for some A > 0. (i) Calculate the volume of this dog-bowl shape as a function of A, leaving your answer in a form that does not involve any trigonometric functions. (ii) For what value of A is the volume of the dog-bowl shape the same as the volume of the conical section that was removed? (b) Let A denote the semi-infinite strip described by a -1 0. Consider the integral

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

6th Edition

ISBN:9781337111348

Author:Bruce Crauder, Benny Evans, Alan Noell

Publisher:Bruce Crauder, Benny Evans, Alan Noell

ChapterA: Appendix

SectionA.2: Geometric Constructions

Problem 10P: A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in...

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