How do you find the volume bounded by #y = 12 ln x#, the x-axis, the y-axis and the line y=12 ln14 revolved about the y-axis?

1 Answer
Eddie
Jun 21, 2016

#1170 pi#

Explanation:

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Working up the y axis, the elemental volume of a small disc of thickness # Delta y# revolved about the y axis will be #Delta V = \pi x^2 Delta y#

where: #x = e^{y/12}# because # y = 12 ln(x)# ..... and so #x^2 = e^{y/6}#

so #V = pi \ int_0^{12 ln(14)} \ e^{y/6} \ dy#
# = 6 \pi [ e^{y/6} ]_0^{12 ln(14)} #
# = 6 pi [ exp((12 ln(14))/6) - 1] #
# = 6 pi [ e^(2 ln(14)) - 1] #
# = 6 pi [ e^( ln(14^2)) - 1] #
# = 6 pi [ 14^2 - 1] = 1170 pi#

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