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Check my work on this please? :-) Surface integral for the parametric equations x = 4 + te\(^t\), y = (t\(^2\) + 1)e\(^t\), 0 ≤ t ≤ 3 Reference: Surface Area = \(\large\int\limits_{a}^{b} 2\pi\ y\ ds \) ds for parametric = \( \large\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\ dt \) So... \[S.A. = \large\int\limits_{a}^{b} 2\pi\ ((t^2+1)e^t) \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}\ dt \] \(dx = te^t\ dt\) \(\large\frac{dx}{dt} = te^t\) \(dy = (t^2+1)e^t+e^t(2t)\ dt \) \(\large\frac{dy}{dt} = e^t(t^2+2t+1) \) \(\large\frac{dy}{dt} = e^t(t+1)^2 \)

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\[\large\int\limits_{0}^{3} 2\pi\ ((t^2+1)e^t) \sqrt{(te^t)^2+(e^t(t+1)^2)^2}\ dt\] \[\large\int\limits_{0}^{3} 2\pi\ ((t^2+1)e^t) \sqrt{e^{2t}(t+1)^2+e^{2t}(t+1)^4}\ dt\] \[\large\int\limits_{0}^{3} 2\pi\ e^{2t}(t^2+1) \sqrt{t^2+2t+2}\ dt\] I'm getting 35833.252388 as the answer, Wolfram says that's ok so I'm looking for a setup error. Help?^%282t%29%29%28t^2%2B1%29sqrt%28t^2%2B2t%2B2%29
I'll have to leave soon unfortunately, but I wanted to get this up before I left because it can take several hours for these hard types of questions to be answered typically :-D Hopefully I don't have any glaring errors >_<
Well actually I hope the error can be found, so the correct answer can be found

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revolving to x-axis yes.
^_^ ty all in advance
Yes sir that is correct, x-axis rotation. That wasn't in the question's specific directions, but it was in the section's directions. Sorry about that. :-/
"...the surface obtained by rotating the given curves about the x-axis."
your dx/dt is wrong. Look at it again.
The 4 goes to 0, yes?
Yes and in total, dx/dt = (t+1)e^t
there is no close solution for integral.
So your answer isn't correct.

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