1. ## volumes rotate about y axis

The region of the curve $\displaystyle y = \frac{e^x}{x}$ bounded by x = 1 and x = 2 and y = 0, is rotated about the y axis. Find the volume of the solid.

i drew the diagram and it doesn't look doable. Would I have to consider volumes by slicing?

thanks

2. It certainly is "doable". The region in the xy-plane is a "quadrilateral with one curved side, vertices at (1, 0), (2, 0), $\displaystyle (2, e^4/2)$ 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 $\displaystyle y= e^x/x$ which is $\displaystyle e^x/x$.

Since this is rotated around the x-axis, the radius of each cylinder is x and the area is $\displaystyle \pi r^2h= \pi x^2(e^x/x)= \pi xe^x$. The volume of each thin cylinder is $\displaystyle \pi x e^x dx$ and the entire volume is given by the integral
$\displaystyle \pi \int_1^2 x e^x dx$.