Setting up an Integral for the area of a surface of revolution

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Homework Help Overview

The problem involves setting up an integral to find the area of the surface generated by rotating the curve defined by the equation y=xe-x over the interval 1 ≤ x ≤ 3 about the y-axis.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the method of setting up the integral, with one attempting to solve for x and another considering substitution to simplify the expression under the square root. Questions arise regarding the correctness of the setup and the presence of an extra term in the integral expression.

Discussion Status

There is an ongoing exploration of the setup of the integral, with some participants providing feedback on the correctness of the expressions used. Clarifications are being sought regarding specific terms in the integral, indicating a productive dialogue about the setup process.

Contextual Notes

Participants mention the need to focus on setting up the integral without evaluating it, which aligns with the homework requirements. There is also a note about potential confusion arising from the notation used in the integral setup.

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Homework Statement



Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve y=xe-x 1=<x=<3 about the y-axis.

Homework Equations



S=integral from a to b x 2pix ds where ds=sqrt(1+(dy/dx)2)dx

The Attempt at a Solution



The first thing I tried to do is solve for the equation in terms of x, and then use the equation above. I figured it makes sense to solve for x since we are rotating the curve about the y-axis. I wasn't able to solve for x, so then I tried to use this method in my textbook where you leave x as it is, and then substitute u for whatever is within the square root sign in such a way that you can eliminate x. I tried to do that, but its turning into a mess since you get 1+(e-x-xe-x)2 underneath the square root and I don't really see how substitution could be used here...any ideas?
 
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All you need to do is set up the integral. Don't worry about trying to evaluate this integral.
 
So does this mean that the way I have set it up is correct? I had a feeling it wasn't right because I couldn't see what steps I'd take next in the event that I had to solve it.
 
Seems to be OK, but I'm a little rusty on these surface area integrals. You have an extra x in what you typed, though, right after b. Did you mean for that to be there?
darkblue said:
S=integral from a to b x 2pix ds where ds=sqrt(1+(dy/dx)2)dx
 
oops, i meant to put a "*" for multiplication.

Thanks for your help!
 

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