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(b) Consider now the function f(x,y) = /R² – x² – y². R. f(x, y) on D is z = f(x, y). Be sure to give some justification fo: (i) Show that the average value of z = (ii) Describe in words the surface z = your description. (iii) Using your work above together with part (a), show that the volume of a ball! o radius R is 4 R3

(b) Consider now the function f(x,y) = /R² – x² – y². R. f(x, y) on D is z = f(x, y). Be sure to give some justification fo: (i) Show that the average value of z = (ii) Describe in words the surface z = your description. (iii) Using your work above together with part (a), show that the volume of a ball! o radius R is 4 R3

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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Please solve 2(b)

Q2. The average value z of z = f(x, y) on a subset D C R² is defined to be 1 z = area(D) JJ, f(7, v) dA (provided the quantities on the RHS are well-defined and finite). In what follows let R> 0 be a fixed constant, let D = {(x, y): x² + y? < R?}, and let f be a continuous function that is defined on D and satisfies f(x, y) > 0 for all (x, y) E D.
Transcribed Image Text:Q2. The average value z of z = f(x, y) on a subset D C R² is defined to be 1 z = area(D) JJ, f(7, v) dA (provided the quantities on the RHS are well-defined and finite). In what follows let R> 0 be a fixed constant, let D = {(x, y): x² + y? < R?}, and let f be a continuous function that is defined on D and satisfies f(x, y) > 0 for all (x, y) E D.
(a) Show that the volume of the solid in R3 that lies below the surface z = f (x, y) and above the region D is equal to the volume of a cylinder of radius R and height z. [Stop and think: Does this assertion make intuitive sense? (You do not have to submit your answer to this question.)] (b) Consider now the function f(x,y) = VR² – x² – y?. f (x, y) on D is z = f (x, y). Be sure to give some justification for {R. (i) Show that the average value of z = (ii) Describe in words the surface z = your description. (iii) Using your work above together with part (a), show that the volume of a ball' of radius R is TR³.
Transcribed Image Text:(a) Show that the volume of the solid in R3 that lies below the surface z = f (x, y) and above the region D is equal to the volume of a cylinder of radius R and height z. [Stop and think: Does this assertion make intuitive sense? (You do not have to submit your answer to this question.)] (b) Consider now the function f(x,y) = VR² – x² – y?. f (x, y) on D is z = f (x, y). Be sure to give some justification for {R. (i) Show that the average value of z = (ii) Describe in words the surface z = your description. (iii) Using your work above together with part (a), show that the volume of a ball' of radius R is TR³.
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