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.

72. The region bounded by f(x) = (x + 1)^(-3/2) and the x-axis on the interval (-1, 1] is revolved about the line y = -1.

Accepted Answer

Below there. Today we're going to 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. Determine the volume generated by revolving the region bounded by H of X is equal to parentheses X + 3 the power of -2. And the X axis on the interval, square brackets of -3.2, about the line Y equals -3. Awesome. So it appears for this particular prom we're asked to determine the volume based on the information that is provided to us by the prom itself. So now that we know that we're ultimately trying to determine what the volume is, let's read off our multiple choice answers to see what our final answer might be. So A is 9 pi divided by 2, B is 3 pi. C is 5 pi and D is does not exist. Awesome. So our first step that we need to take in order to solve this problem is we need to take a moment to understand the geometry of this particular prom. So with that in mind, let us recall that based on the information that's provided to us by the problem, the region lies above the x axis, and it is bounded between X is equal to -3 and X is equal to 2. And that we are to revolve around Y equals -3. So Y equals -3 is what we're revolving around, and this will be 3 units below the X axis. So because we're revolving around Y equals -3, so that means it'll be 3 units below the X axis. So that means as a result that for each X. The distance from the curve to the axis of rotation will be R of X is equal to H X minus -3. Which is equal to H X + 3. So this is the outer radius of the washers. Now turning our attention to the inner radius, the inner radius is from the x axis to Y equals -3, which is a constant which will state, and let's take a moment. When I say R of X, let's specifically note that this is a capital R. and when we're referring to our inner radius, so our outer radius. Is the capital R of X, and let's say that our inner radius. Which will be lowercase r of X. So once again, our inner radius is from is from our x axis 2 Y equals -3, which is a constant. So we could say that lowercase r of X is equal to 0 minus -3, which is equal to 3. So that means that we are using the washer method, as we should recall, and because we are using the washer method, as we should recall a note, our formula will become as follows the volume is equal to, so once again, our volume will be equal to pi, multiplied by the integral from -3 to 2. Of square brackets of parenthesis of H X plus 3 2 minus 3 to the power of 2. DX. So our next step now is we need to simplify the expression inside of our square brackets. So we need to focus on parentheses H X + 3 to the power 2 minus 9. So when we simplify what's inside of our square brackets, we will obtain the following. So parenthesis of H X + 32 minus 9. So we now need to substitute, so let's make a note, we now need to substitute H of X is equal to 1 divided by parentheses X plus 3 to 2, and when we substitute, we will obtain the following. We will state that H of X is We're going to state that H of X + 3 is equal to 1 divided by parentheses X + 32 + 3, which will mean that parentheses H X + 3, to the power 2 will be equal to parentheses 1 divided by parentheses X + 32 + 3. All to the power 2. So now we need to compute this particular expression. So our second step will be, in order to compute this expression, we need to expand. Are parenthesis of H X + 32. So with that in mind, since we need to expand that expression, let us denote and let's make a note in blue, let us denote that you will be equal to 1 divided by parentheses X + 32. So that will mean we can go ahead and write that parentheses H X + 3 to 2 will be equal to parentheses U + 3 to the power 2, which will be equal to U 2 + 6 U + 9. So, we can go ahead and write that parentheses H X + 32 minus 9 will be equal to U 2 + 6 U. Which will be equal to parentheses 1 divided by parentheses X + 32. All to the power 2, plus 6 multiplied by 1 divided by parentheses X + 3 to the power of 2. Which will be equal to 1 divided by parentheses X + 3 to the power of 4 + 1 and it's going to be plus 6 divided by parentheses X plus 3 to the power of 2. Fantastic. So, Our third step now, so we're going on to our 3rd step, is we need to set up our integral. So when we set up our integral, we will find that the volume is equal to pi multiplied by the integral from -3 to 2 of parentheses of 1 divided by parentheses X + 3. To the power of 4 plus 6 divided by parentheses X plus 3 to the power of 2 D X. Awesome. So now we're moving on to our fourth step. So our 4th step is we need to determine the change of variable. So with that in mind, let us say that U is equal to x + 3, so that will mean that DX is equal to DU. And we need to change our limits. So when X is equal to -3, that will mean that U is equal to 0, and when X is equal to 2, that will mean that U is equal to 5. So that would mean that we could go ahead and write that V the volume is equal to pi multiplied by the integral from U equals 0 to 51 divided by 4 plus 6 divided by U 2 in parentheses. Fantastic. Our 5th step now is we need to check for improper behavior. So with that in mind, we need to be able to recognize, or rather notice, that the lower limit is U is equal to 0. And both terms are 1 divided by U to the power of 4 and 1 divided by U to the power of 6, or rather 6 divided by you to the 2. So that will mean that it will diverge, which I'll simply call D. so it will diverges. So that will mean that you approaches 0 from the right. That's what that positive symbol and the power means. So you is going to be approaching 0 from the right. So specifically, That will mean that since it's diverging, you can write that the integral from 0 to 5 of 1 divided by U to the power of 4, D is equal to the limit. As E approaches 0 from the right of the integral of E to 5, so it's going to be the integral from E to 5 of U to the power of negative 4DU, which will be equal to the limit as E approaches 0 from the right of square brackets of U to the power of -3 divided by -3, evaluated from. E to 5. Which will mean that this is equal to the limit, as E approaches 0 from the right of parentheses 1 divided by 3, multiplied by E to the power of 3 minus 1 divided by 375, which will be ultimately equal to positive infinity. So with that in mind, the integral, as we should recall a note, diverges doing, so it's going to diverge. So since this integral diverges, and this is due to the singularity at X is equal to -3. For and this is meaning that u is equal to 0. So because the interval diverges due to singularity, that will mean without a doubt that the volume V does not exist. So we can simply write D and E, so you'll probably make C. That the acronym DNE is used for does not exist. And that's our final answer. And this is because the integral diverges due to singularity and because it diverges, that means that the volume for this particular problem does not exist. So looking at our multiple choice answers, the final answer, without a doubt, will have to be multiple choice answer letter D does not exist. 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.
72. The region bounded by f(x) = (x + 1)^(-3/2) and the x-axis on the interval (-1, 1] is revolved about the line y = -1.

Key Concepts

Volume of Solids of Revolution

Improper Integrals and Convergence

Function Behavior and Domain Restrictions

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