Posted by **Sarah** on Tuesday, March 8, 2011 at 8:24am.

Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.

y = ln x, y = 4, y = 5, x = 0

Find the volume V of this solid.

Help!!! Thank you in advance :(

- Calculus II -
**MathMate**, Tuesday, March 8, 2011 at 9:44am
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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