6.3 Integral Applications
Question #2 (Medium): Volume by Cylindrical Shells Rotated About the X-Axis
First helpful step is to graph the given function and lines.
Recalling the volume calculation using cylindrical shells: ∫
Given function defines the interval[ ]for integral and which curve denotes the of the
cylindrical shells. If this function is given in the form of , when the rotation is about the -axis,
then rearrange the function to get the inverse . Then the volume is written as:
If the function describing the height is given in the form of , no rearrangement is necessary,
and so it can be used directly for the integral.
Using cylindrical shells find the volume obtained by rotating the region bound by the curves about the -
√ , ,
By graphing the function and the lines, i