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.

75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-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 by revolving the region bounded by F of X is equal to parenthesis of 3 minus X to the power of negative 1/2. And the x axis on square bracket of 0.3 parenthesis about the y axis. Awesome. So it appears that we're asked to take the information that is provided to us by the prom itself and we're asked to find the volume of this particular solid. So now that we know that we're ultimately trying to solve for the volume of this solid, let's 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 8 pi, B is 8 pi multiplied by the square root of 3, C is 2 pi multiplied by the square root of 3, and finally D is pi multiplied by the square root of 3. Fantastic. So our first step that we need to take in order to solve this problem is we need to recall and use the shell method. With that in mind, we also need to recall and note that since we are revolving around the y axis, that means that the region is given in terms of F of X. That means that the shell method is the most natural choice. So with that in mind, the shell method formula, as we should recall, states that the volume V is equal to 2 pi multiplied by the integral from A to B of X multiplied by F of XDX. Where in this case for here in this particular prom, A will be equal to 0 and B will be equal to 3, and F of X will be equal to parentheses of 3 minus X to the power of -12. So, therefore we can go ahead and write that V is equal to 2 pi multiplied by the intercal from 0 to 3 of X multiplied by parentheses 3 minus X to the power of negative 12 DX. fantastic. Our second step now is we need to perform substitution. So let us state that U is equal to 3 minus X. So that will mean that DU is equal to negative d X, and that will also mean that X is equal to 3 minus U. Fantastic. So now we need to change our bounds. So when X is equal to 0, that will mean that U is equal to 3, and when X is equal to 3, that will mean that U is equal to 0. So that means therefore, that VR volume will be equal to 2 pi multiplied by the integral from x equals 0 to 3 of X multiplied by parentheses 3 minus x the power of -12 DX. Which will be equal to 2 pi multiplied by the integral from U equals 3 to 03 minus U multiplied by U to the power of negative 1/2 multiplied by negative DU fantastic. And this will be equal to 2 multiplied by the integral from 0 to 3 of parentheses 3 minus U multiplied by U to the power of negative 12 DU. Fantastic. So now we need to expand, so we need to take -3 minus U multiplied by U to the power of -12, which will be equal to 33 multiplied by U to the power of -12 minus u to the power of 1/2. Fantastic. So with that in mind, after our expansion, the integral will become V is equal to 2 multiplied by the interval from 0 to 3 of 3 multiplied by the power of negative 1/2 minusu of 12 DU. Fantastic. So now we need to perform our integration, and that will be our third step. So we need to, as firstly, since we're integrating, we need to take the integral from 0 to 3 of 3 or 3 multiplied byu to the power of negative 1/2 DU, which is equal to 3 multiplied by square brackets of 2 multiplied by U to the power of 1/2. Evaluated from 0 to 3, which is equal to 3 multiplied by 2 multiplied by the square root of 3, which simplifies 2, which meaning it will equal to. 6 multiplied by the square root of 3. And the integral from 0 to 3 of U to the power of 12 DU will be equal to square brackets of 2 divided by 3 multiplied by U to the power of 3 divided by 2, evaluated from 0 to 3, which will be equal to 2 divided by 3 multiplied by parentheses 3 to the power of 3 divided by 2. Which, when we simplify, it will be ultimately equal to 2 multiplied by the square root of 3. Fantastic. So, with that in mind, Moving right along here. So that will mean we now need to subtract, so we need to take 6 multiplied by the square root of 3, minus 2 multiplied by the square root of 3, which will be equal to 4 multiplied by the square root of 3. And now we need to solve for the volume and in order to do that, in order to solve for the volume, we need to multiply by 2 pi. So the volume V is equal to 2 pi multiplied by 4 multiplied by the square root of 3, which will be equal to 8 pi multiplied by the square root of 3, and that is our final answer. Hooray, we did it. So looking at our multiple choice answers, our final answer without a doubt will have to be multiple choice answer letter B, and once again we we can't forget our units. So our final answer will be 8 pi multiplied by the square root of 3 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.75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.

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65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.
75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.