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#1 2007-04-15 05:27:05

canny
Member
Registered: 2007-04-14
Posts: 1

Calculus Problem

Let R be the region bounded by y=lnx, the x-axis, and the line x=e.
(a) Find the volume of the solid generated when R is rotated about the x-axis.
(b) Find the volume of the solid formed with R as base and such that cross sections perpendicular to the y-axis are squares.

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#2 2007-04-15 06:16:28

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Calculus Problem

Last edited by Stanley_Marsh (2007-04-15 06:27:24)


Numbers are the essence of the Universe

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#3 2007-04-15 07:24:03

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: Calculus Problem

The formula for volume of solid of revolution is

For for region R from x = 1 to x = e, it’s

So integrate [ln(x)][sup]2[/sup]. Hint: Integrate [ln(x)][ln(x)] by parts.

For the second part, shift the curve by e units to the left, so R is now y = ln(x+e) from x = 1−e to x = 0. (The new graph cuts the y-axis at (0,1).) But you now need to integrate along the y-axis, so you must express x in terms of y: x = e[sup]y[/sup] − e.

For an element of increase from y to yy along the y-axis, the element of volume sliced out has base area xΔy cross-sectional area x[sup]2[/sup]; ∴ the element of volume is ΔV = x[sup]2[/sup]Δy.

Last edited by JaneFairfax (2007-04-17 08:48:20)

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#4 2007-04-17 08:08:50

canny
Member
Registered: 2007-04-14
Posts: 1

Re: Calculus Problem

thanks

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