Calculate volume with respect to y

**kallekall** · July 29th 2009, 04:49 AM

$y=\ln{x} , 1\leq x\leq e$

Calculate the volume between y and the x-axis with:

a) integration with respect to x.

b)integration with respect to y.

I did a) without any problem, but I don't get how to do with b).

The correct answer are $2\pi \int_{0}^{1} y(e-e^y) dy$

How should I think? I get that I should change to equation from $y= \ln{x} , 1\leq x\leq e$ to $x=e^y , 0\leq y\leq 1$ and that the but then I got no idea at all...

**songoku** · July 29th 2009, 05:43 AM

Use another formula to find volume :
$V = 2\pi \int (y-k) x dy$ , k is the axis of rotation

Because the integral is (dy), the part that are being integrated is between the curve and y-axis.
First, you find the volume of the section that are bounded by y-axis and x = e, then find the volume of the section that are bounded by the curve and y-axis.

Find out which one has bigger volume then substract it with the smaller one.

Sorry, it's difficult for me to explain it with words....

**HallsofIvy** · July 29th 2009, 05:49 AM

Originally Posted by kallekall

$y=\ln{x} , 1\leq x\leq e$

Calculate the volume between y and the x-axis with:

Volume between y and the x-axis? Are you assuming rotation around the x-axis? It would be nice to tell us!

a) integration with respect to x.

b)integration with respect to y.

I did a) without any problem, but I don't get how to do with b).

The correct answer are $2\pi \int_{0}^{1} y(e-e^y) dy$

How should I think? I get that I should change to equation from $y= \ln{x} , 1\leq x\leq e$ to $x=e^y , 0\leq y\leq 1$ and that the but then I got no idea at all...

**DeMath** · July 29th 2009, 06:02 AM

Also look carefully this picture

**kallekall** · July 31st 2009, 05:52 AM

Originally Posted by HallsofIvy

Volume between y and the x-axis? Are you assuming rotation around the x-axis? It would be nice to tell us!

Sorry songoku and HallsofIvy, I was a bit unclear there... I'm suppose to rotate around the x-axis.

Later found this in the tips-section:

("Lösning" means solution, if anyone were wondering

)

**songoku** · July 31st 2009, 06:10 AM

1. Find the volume when area bounded by x = e and y-axis is rotated around the x-axis
2. Find the volume when area bounded by the curve and y-axis is rotated around the x-axis

Use formula : $<br /> V = 2\pi \int (y-k) x dy<br />$

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