# volumes rotate about y axis

• April 27th 2010, 02:57 AM
differentiate
The region of the curve $y = \frac{e^x}{x}$ bounded by x = 1 and x = 2 and y = 0, is rotated about the y axis. Find the volume of the solid.
It certainly is "doable". The region in the xy-plane is a "quadrilateral with one curved side, vertices at (1, 0), (2, 0), $(2, e^4/2)$ and (1, e). There would be a problem with doing it as "washers" since there would be a change of formula at (1, e).
If you "slice" it parallel to the y-axis, rotating around the y-axis, each "slice" would be a cylinder, of thickness dx, and height the distance from y= 0 to $y= e^x/x$ which is $e^x/x$.
Since this is rotated around the x-axis, the radius of each cylinder is x and the area is $\pi r^2h= \pi x^2(e^x/x)= \pi xe^x$. The volume of each thin cylinder is $\pi x e^x dx$ and the entire volume is given by the integral
$\pi \int_1^2 x e^x dx$.