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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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 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.
(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<y sz +1 and a 2 0. Consider the integral
r°(y – x)"
dz dy,
I(a, n) = L1+1)2a+2
where a > -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.
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