Question

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 :(

Answer 1

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.