Question

Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.
y = ln(x), y = 1, y = 5, x = 0
Find the volume V of this solid.

Answer 1

ok this time the cross-sections are horizontal because we are rotating about the y-axis
Since we are rotating the shape of each cross-section is a circle
Area of circle is:
\[\pi*r^{2}\]
the radius can be represented by the x-distance the line ln(x) is away from y-axis
\[y= \ln x \rightarrow x = e^{y}\]
so \[r =e^{y}\]
Integral looks like this
\[V = \pi \int\limits_{1}^{5}(e^{y})^{2} dy\]