2. Consider part of the graph of y = ! for x > 1. By rotating the graph about the x-axis, the generated unbounded solid region is called the Gabriel's Horn (Archangel Gabriel who blows the horn to announce Judgment Day), or Torricelli's trumpet (named after an Italian physicist and mathematician Evangelista Torricelli 1608-1647). (a) Find the volume of the Gabriel's Horn. (b) The surface area of an object generated by rotating part of the non-negative graph of y = f(x) along the x-axis for a < x < b is given by A = | f(x)/1+ (f'(x))²dx. Show that the Gabriel's Horn has an infinite surface area, although it has a finite volume (obtained in (a) above). (This is known as the Painter's Paradox, that a finite volume of paint can 'filled-up' the Horn, but yet, never enough to paint the infinite inner surface).
2. Consider part of the graph of y = ! for x > 1. By rotating the graph about the x-axis, the generated unbounded solid region is called the Gabriel's Horn (Archangel Gabriel who blows the horn to announce Judgment Day), or Torricelli's trumpet (named after an Italian physicist and mathematician Evangelista Torricelli 1608-1647). (a) Find the volume of the Gabriel's Horn. (b) The surface area of an object generated by rotating part of the non-negative graph of y = f(x) along the x-axis for a < x < b is given by A = | f(x)/1+ (f'(x))²dx. Show that the Gabriel's Horn has an infinite surface area, although it has a finite volume (obtained in (a) above). (This is known as the Painter's Paradox, that a finite volume of paint can 'filled-up' the Horn, but yet, never enough to paint the infinite inner surface).
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:2. Consider part of the graph of y = ! for x > 1. By rotating the graph about the x-axis,
the generated unbounded solid region is called the Gabriel's Horn (Archangel Gabriel
who blows the horn to announce Judgment Day), or Torricelli's trumpet (named after
an Italian physicist and mathematician Evangelista Torricelli 1608-1647).
(a) Find the volume of the Gabriel's Horn.
(b) The surface area of an object generated by rotating part of the non-negative graph
of y = f(x) along the x-axis for a <x < b is given by
"/ $(x)/1+ (f'(x))*dx.
A =
27
Show that the Gabriel's Horn has an infinite surface area, although it has a finite
volume (obtained in (a) above). (This is known as the Painter's Paradox, that a
finite volume of paint can 'filled-up' the Horn, but yet, never enough to paint the
infinite inner surface).
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