(a) Show that the volume of the solid in R 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.)]
(a) Show that the volume of the solid in R 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.)]
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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Please solve this question correctly
![The average value z of z = f(x, y) on a subset D C R? is defined to be
1
/, f(x, y) dA
area(D)
(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? < R²}, and let f be a
continuous function that is defined on D and satisfies f(x, y) 2 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) =
V
R² – x² – y².
(i) Show that the average value of z = f(x, y) on D is z =R.
(ii) Describe in words the surface z = f(x, y). Be sure to give some justification for
your description.
(iii) Using your work above together with part (a), show that the volume of a ball of
radius R is TR.
(Note: In c. 225 BCE, Archimedes showed that the volume of a ball of radius R
is equal to nR³ by showing that it is equal to two-thirds the volume of the smallest
cylinder that surrounds it. He did so using an ingenious argument that involved an early
form of integral calculus, almost 2000 years before the "official" discovery of calculus by
Newton and Leibniz. Our approach above is very close in spirit (though not identical
to) Archimedes's method.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc62bdeb3-ddfb-4b0f-a88a-41d3e9d56694%2Ff85bf691-fcff-45d0-acbd-9a65fb59499f%2Fo0stv4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The average value z of z = f(x, y) on a subset D C R? is defined to be
1
/, f(x, y) dA
area(D)
(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? < R²}, and let f be a
continuous function that is defined on D and satisfies f(x, y) 2 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) =
V
R² – x² – y².
(i) Show that the average value of z = f(x, y) on D is z =R.
(ii) Describe in words the surface z = f(x, y). Be sure to give some justification for
your description.
(iii) Using your work above together with part (a), show that the volume of a ball of
radius R is TR.
(Note: In c. 225 BCE, Archimedes showed that the volume of a ball of radius R
is equal to nR³ by showing that it is equal to two-thirds the volume of the smallest
cylinder that surrounds it. He did so using an ingenious argument that involved an early
form of integral calculus, almost 2000 years before the "official" discovery of calculus by
Newton and Leibniz. Our approach above is very close in spirit (though not identical
to) Archimedes's method.]
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