Posted by Sarah on Tuesday, March 8, 2011 at 8:24am.
This problem can be easily solved using the disk method.
Horizontal disks are used, with slices of thickness dy.
We will integrate from y=4 to y=5.
Each disk has a volume of πr(y)²dy.
where the radius is a function of y.
Since y=ln(x), its inverse relation is x=e^y.
Integrate for y=4 to 5 of
V=∫π(e^y)²dy
=π∫e^(2y)dy
=π(1/2)e^(2y)
Evaluate between 4 and 5 gives
V=(π/2)(e^(2*5)-e^(2*4))
=29917 (approx.)
Check:
The average radius is between e^4 and e^5=101.5
Volume = 32400 approx. > 29917
Since the curve ln(x) is concave up, the actual volume should be a little less than the approximation. So the calculated volume should be correct.
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