Question about solids of revolution

[Nootropic]

Sorry for creating a second thread, but this problem is eating at my head, and I should have put in here in the first place, but anyhow, here are my posts on the thread which was (well, still is...) in the physics and mathematics.

 

so I've been doing solids of revolutions on my own in calculus and I've come across an AP question I'm not sure about. It goes a little something like this: "Let R be the region enclosed by the graphs of y = e^x and y = (x-1)^2, and the line x = 1." Now the first two questions were not of difficulty (find the area of R and find the volume of the solid if R is revolved about the x-axis). Now the final questions asks to find the volume of the slid when R is revolved about the y-axis. Now this is relatively easy to do with the "shell method", but I'm curious if there's a way to do it with "washers" about the y-axis that would be more efficient.

 

Gah, don't stay up till 7 in the morning partying. I meant to place this in the homework and projects section, but I am apparently still exhausted and absentminded (though the latter's mostly a constant even in the absence of exhaustion). Anyhow, feel free to move it!

 

I suppose I could throw out the work I've done on finding a way to revolve around it around the y-axis. The two curves intersect at at x = 0 or when y = 1 (verify it if you want). So one of the limits of integration has to be y = 1, and solving for y gives ln y = x and x = (y)^1/2 + 1, however, the problem I have is what is the other limit of integration? And another oddity, when solved for y, the curves should intersect when y = 1, but ln (1) = 0 and 1^(1/2) + 1 = 2. I don't think I made a mistake, but I might have and am probably not seeing it...back to the drawing board.

 

There appears to be a solution on this page (us ctrl + f and type in "y = e^x to find quickly), but, first of all, this is a strange page and why are there derivatives involved in revolving this surface? It begs for an explanation. Previous Questions