Let
- Determine the slope of the graph of
at the point with
-coordinate 1.
- Find the volume of the solid of revolution formed by rotating the region between the graph of
and the interval
about the
-axis.
- To take this derivative, using logarithmic differentiation will be easier,
Then differentiating both sides we have,
So, to find the slope at the point with
we evaluate,
- First, the integral to compute the volume of the solid of revolution is,
To evaluate this we use the partial fraction decomposition,
This gives us the equation
Evaluating at
,
, and
we obtain
Therefore, we have