Volumes
Find the volume of the solid generated by revolv...

Question

Volumes

Find the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4/x³ and the lines x = 1 and y = 1/2 about

c. the line x = 2

Accepted Answer

Hello there, today we are 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. Find the volume of the solid generated by revolving the region bounded by y is equal to 8 divided by x. The vertical line x is equal to 1, and the horizontal line y is equal to 2. About the vertical line x is equal to 0. The y axis. So it appears for this particular prong we are asked to use all the information that is provided to us and we're asked to use this information to find the volume of this solid. So now that we know that we're ultimately trying to solve for the volume of this specific solid. We need to be able to look at the information that's provided to us, and we need to recognize, based on our prior knowledge that we can recall and use the shell method formula because this is for cylindrical shells for revolution about a vertical line. X is equal to A. So that means based on our prior knowledge we could state that the equation for the, or rather the formula for the volume V, so the volume V will be equal to 2 high multiplied by the integral evaluated from A to B of parentheses of the radius R multiplied by parentheses of H, the height, multiplied by dx. And we need to note that the radius R. Is equal to the absolute value of x minus a. And the height H is equal to F X minus G X. Fantastic. So with this in mind, we can now apply the shell method formula to the region bounded by Y is equal to 8 divided by X. And the vertical line at X is equal to 1, and the horizontal line Y is equal to 2, and we need to note that it is revolved, which I'll call capital R in quotes, and it's revolved about X is equal to 0. So now, at this point, we need to find the right boundary by setting 8 divided by x is equal to 2. So that will mean when we rearrange this expression to isolate and solve for x, we will find that x is simply equal to 4, and we just need to use a little bit of algebra to rearrange to get x all by itself on one side of our expression. So that will mean that our radius R will be equal to the absolute value of x minus 0 in absolute value signs, which will be equal to the absolute value of x, which will be equal to x, and this will be on the closed interval square brackets of 1.4, fantastic. And we need to note that 8 divided by X is greater than or equal to 2 on our closed interval of square brackets of 1.4. Fantastic. So with that in mind, we could therefore state without a doubt that the height H is equal to 8 divided by X minus 2. So now we could substitute these variables back into our shell method formula to solve for the volume. So we could say that the volume V is equal to 2 pi multiplied by the integral evaluated from 1 to 4 of x multiplied by parentheses of 8 divided by x minus 2. DX Which you go ahead and write that the volume V is equal to 2 pi multiplied by the integral evaluated from 1 to 4 parentheses of 8 minus 2 XDX. Which we can go ahead and write that V is equal to 2 pi multiplied by parentheses of. 8 x minus x 2. And we need to note that this will be evaluated from 1 to 4. Thus we could write that v is equal to 2 pi multiplied by square brackets of parentheses of 8 multiplied by 4 minus 42 minus 8 multiplied by 1 minus 12. Which will mean when we plug this expression into our calculator and we write our final answer in its most simplified form. We will find that the. Will be equal to 18 pi. And that's it. That is our final answer. But before we go any further, we need to note that we have to write our units, and since we're dealing with volume, we need to note that this will be in cubic units. So that's it. That's our final answer. So in case you're a little confused about how we got 18 pi, let us note here, I'll write a little hint here. So going back to our volume expression in our square brackets, we can note that. This will simplify to V is equal to 2 pi multiplied by parentheses of 32 minus 16. -8 + 1, which will mean that V is equal to 18 when we plug that expression into our calculator. So you just need to take it step by step. Don't rush it and just, and you will solve it, I promise. Just breathe and you got this. So, with that said, we've officially solved for this problem. Hooray. So without a doubt, our final answer once again has to be 18 pi cubic units, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

VolumesFind the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4/x³ and the lines x = 1 and y = 1/2 aboutc. the line x = 2

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Volumes
Find the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4/x³ and the lines x = 1 and y = 1/2 about
c. the line x = 2