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Numerade Educator

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Problem 41 Hard Difficulty

The region bounded by the curve $ y = \frac{1}{(1 + e^{-x})} $, the x and y-axes, and the line $ x = 10 $ is rotated about the x-axis. Use Simpson's Rule with $ n = 10 $ to estimate the volume of the resulting solid.

Answer

$V \approx \pi I_{1} \approx 27.7$ or 28 cubic units.

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Video Transcript

Okay, so this question wants us to estimate the volume of the salad formed by revolving this bounded region around the X axis. So if it's bounded by the red curve, the green line, the blue line and the purple line well, a region is just this thing here and well, what is that? Well, that's just the area underneath the graph of the red curve between zero and 10. So f of X equals one over one plus eat of the minus X, and our radius of our show is just equal to F of X. So our volume we're using the disc method is just pi times the integral from 0 to 10 of f of X squared DX, which in this case is pie times the Inter girlfriend 0 to 10 of one over one plus e to the minus x squared D X. So now we have our expression for the volume and we need to use Simpson's rule to approximate it. So it says use and equals time. So Delta X is we're ending at 10 starting at zero with 10 to sub intervals. So Delta X is one. So that means that s a turn is approximately equal to the volume and Essam 10 equals well, Delta X is one so we get one over three times F zero squared plus four f of one squared plus two f of two squared plus all the way up to F of 10 squared with a factor of pi Tek Don because pie comes along with the volume integral. So if we evaluate this, we see that the volume there's approximately equal to 28 and again First we found our region to set up our volume integral right here and then to find the integral. We just used Simpsons Rule being extra careful to use our factor of pie in front and two square every one of our F values like it is in the integral.