65-76. Volumes Find the volume of the described solid of revol...

Question

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

69. The region bounded by f(x) = 1/√(x ln x) and the x-axis on the interval [e, ∞) is revolved about the x-axis.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the volume of the solid generated when the region is bounded by F of X is equal to 1 divided by x to the power of 1/2 multiplied by parentheses the natural log of X, the power of 3 divided by 2. And the X axis on the interval, square bracket E to the 2 infinity parenthesis. Is revolved about the x axis. Awesome. So it appears for this particular problem we're asked to take the information that is provided to us and we're asked to use that information to help us find the volume of this particular solid. So now that we know that we're trying to solve for the volume, let us read off our multiple choice answers to see what our final answer might be, noting that they're all in the same units of cubic units. So A is pi divided by 4, B is pi divided by 8, C is pi divided by 2, and D is pi divided by 16. Awesome. So our first step that we need to take in order to solve this problem is we need to recall and use the disk method. So with that in mind, we need to recall what the volume formula is. And as we should recall, the volume formula states that V is equal to pi multiplied by the integral from A to B of square bracket. Or square brackets of F of X to the power to DX. So now we need to substitute in F of X, and when we do, we will find that F of X to the power of 2 will be equal to parentheses 1 divided by the square root of X, multiplied by parentheses the natural log of X to the power of 3 divided by 2. Fantastic. And we need to take this all to the power 2, so we need to take 1 divided by the square root of X multiplied by the natural log of X to the power of 3 divided by 2 to the power 2. And when we do, we will find that it's simply equal to 1 divided by X multiplied by parenthesis, the natural log of X to the power of 3. Fantastic. So therefore, our volume formula will become V is equal to pi multiplied by the integral from E to the power of 2 to infinity of 1 divided by X, multiplied by parentheses the natural log of X to the power of 3D X. Fantastic. So, our next The point of action that we need to take is, or rather, our second step that we need to take is we need to perform substitution. So let us state that U is equal to the natural log of X. So that will mean that DU is equal to 1 divided by XD X. And we need to note that when X is equal to E to the power 2, You will be equal to 2, and as X approaches positive infinity. That will mean that you approaches positive infinity as well, so they both approach positive infinity X and U. So therefore, with this in mind, we can go ahead and write that 1 divided by X multiplied by parenthesis of the natural log of X. To the power of 3, DX is equal to 1 divided by U to the power of 3 multiplied by 1 divided by XDX which is equal to 1 divided by U to the power of 3DU. So with that in mind, our integral will become V is equal to pi multiplied by the integral of U is equal to 2. So it's gonna be pi multiplied by our integral from U is equal to 2 to positive infinity of 1 divided by U to the power of 3D. Fantastic. So that will mean our 3rd step, this is our 3rd and final step. We need to evaluate the integral now. So we have the integral from 2 to positive infinity of 1 divided by U to the power of 3, D U. Which is equal to square brackets of U to the power of -2 divided by -2, evaluated from 2 to infinity, which is equal to 0. So it's gonna be equal to 0 minus parentheses 1 divided by -2 multiplied by 4, which will be equal to 1/8 or 1 divided by 8. So that will mean that our volume V will be equal to pi multiplied by 1/8 or 1 divided by 8, which will give us an answer that is equal to pi divided by 8. And that's it. That is our final answer for the volume. Hooray, we did it, and don't forget that our units are in cubic units, and that's it. We've officially solved for this problem. So looking at our multiple choice answers, our final answer, without a doubt, will have to be multiple choice answer letter B, which once again is pi divided by 8 cubic units. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.69. The region bounded by f(x) = 1/√(x ln x) and the x-axis on the interval [e, ∞) is revolved about the x-axis.

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65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
69. The region bounded by f(x) = 1/√(x ln x) and the x-axis on the interval [e, ∞) is revolved about the x-axis.