Which of the following integrals computes the volume of the solid formed by revo

Question

Which of the following integrals computes the volume of the solid formed by revolving the region bounded by the graph of y = Ln(x), the x-axis, and the line x = 3 about the line x = 3?

a) https://www.connexus.com/content/media/308520-6292011-82942-AM-245436193/webdav/res0094/educator_apcalc/Testpix/v7/7_05_33a.gif

b) https://www.connexus.com/content/media/308520-6292011-82942-AM-245436193/webdav/res0094/educator_apcalc/Testpix/v7/7_05_33b.gif

c) https://www.connexus.com/content/media/308520-6292011-82942-AM-245436193/webdav/res0094/educator_apcalc/Testpix/v7/7_05_33aa.gif

d) https://www.connexus.com/content/media/308520-6292011-82942-AM-245436193/webdav/res0094/educator_apcalc/Testpix/v7/7_05_33d.gif

Explanation / Answer

V = integral of 2rf(x)dx....limit from 0 to 3
r = x-3
f(x) = lnx
V = integral of 2(x-3)lnxdx
V = integral of 2(xlnx-3lnx)dx
integrating wrt x
V = 2[x(xlnx-x) - xlnx + x - 3xlnx]
V = 2[x^2lnx - x^2 - 4xlnx + x]
apply limits
V = 2[9ln3 - 9 - 12ln3 + 3]
V = 2[-3ln3 - 6]
since it cannot be negative
V = 6(ln3 + 1)

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