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Numerade Educator

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Problem 38 Hard Difficulty

Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

$ y = x $ , $ y = xe^{1 - \frac{x}{2}} $ ; about $ y = 3 $

Answer

$V=\pi \int_{0}^{2}(3-x)^{2}-\left(3-x e^{1-x / 2}\right)^{2} d x=24 e-\frac{142}{3}-2 e^{2}$

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Video Transcript

she need. Yeah. Were given curves and align and rescues computer out of her system to find the exact volume of the salt obtained by rotating the region bounded by these curves about this line. The curves are Y equals X and Y equals X times E. To the one minus X over two. See Oh it's offenders take. That's why so don't be afraid. In the line we rotate is about the line Y equals three. It might help if you want to graph these curves on your own along with the axis of rotation. If you do so, you will see that cross section is a washer. The washer has an outer radius which is why equals three minus our bottom function, which is Y equals X. They just need food now. And the inner radius of the washer is three minus the top function, which is X times E. To the one minus X. Over two guys fucking adult home on it. He's been an employee who was in the wrong now. It follows from the washer method at the volume of the solid is given by pi times the integral from, well, if you look at this function, let's find where they intersect. So for example, and Y equals X and Y equals X times eat the one minus X over two sen seven And they're not a full taquito either. We have X times one minus E. To the one minus X over two equals zero. And so we have this zeros X equals zero war E to the one minus X over two equals one. And therefore one minus X over two is the natural log of one, which is zero and therefore X is equal to two. Now, if you plug in We get the zero And the point uh to to although you really don't have to do this, all we need to know is that X ranges from 0-2 are integral. And we have our outer radius three minus X squared minus the inner radius three minus X times E. To the one minus x over two squared dx. Yeah. And if you use a computer algebra system to evaluate this lengthy integral, The exact dancer is 24 e. Well, you shouldn't be able to buy Kane -142/3 minus two E squared when, So this is the exact value of the volume. Yeah, it's about use person.