can someone please explain 5b to me (1 Viewer)

[Cherrybomb56]

Can someone explain 5b, I got 5a but don't understand 5b.

 

[tickboom]

This is a classic volumes of revolution question. Have you learned about the method of slices or the method of cylindrical shells? You could probably use either here ... It will ultimately result in an integration, but the key is understanding how to visualise what happens when you rotate the shape around the axis to then find the components of the integral. I'll be happy to spin up a video on this if you think it will help.

 

[username_2]

Can someone confirm the solution:

 

[Cherrybomb56]

This is a classic volumes of revolution question. Have you learned about the method of slices or the method of cylindrical shells? You could probably use either here ... It will ultimately result in an integration, but the key is understanding how to visualise what happens when you rotate the shape around the axis to then find the components of the integral. I'll be happy to spin up a video on this if you think it will help.

can you please send a video.

 

[tickboom]

can you please send a video.

Can do! Hopefully I can get it done later tonight, but Handmaid's Tale is on so we'll see how I go ... Otherwise perhaps tomorrow night.

But looking at username_2's solution, I'd say that all looks ok, except for the very last line where it looks like the π has inexplicably dropped off. I think the final solution should be approx 0.84 ... (in other words, the 0.27 multiplied by π) ...

But I think the key to understanding this question is working out why the volume is the result of rotating x=1/2 around the y-axis and then subtracting the rotation of the curve, which hopefully will become clear when I record a video and draw up the diagram.

 

[tickboom]

Can do! Hopefully I can get it done later tonight, but Handmaid's Tale is on so we'll see how I go ... Otherwise perhaps tomorrow night.

But looking at username_2's solution, I'd say that all looks ok, except for the very last line where it looks like the π has inexplicably dropped off. I think the final solution should be approx 0.84 ... (in other words, the 0.27 multiplied by π) ...

But I think the key to understanding this question is working out why the volume is the result of rotating x=1/2 around the y-axis and then subtracting the rotation of the curve, which hopefully will become clear when I record a video and draw up the diagram.

Ok video is ready to rock! Enjoy!

 

[Cherrybomb56]

Ok video is ready to rock! Enjoy!

Thank you. That was helpful.

 

[CM_Tutor]

Does whatever book this comes from give the solution as

The wording of the question says that "the corresponding area enclosed between the curve and the -axis is rotated about the -axis" and this sounds to me like the area bounded by the curve and the -axis between and is the volume being sought, rather than rotating the region bounded by the curve, the line , and the coordinates axes. If it is the region whose area was found in part (a) that is rotated, then the first answer is sought... if the region rotated is the different (and smaller) region bounded the curve, the -axis, and , then the question is easier as the subtraction from the cylinder is not needed, and the latter answer above applies.

 

[tickboom]

Does whatever book this comes from give the solution as

The wording of the question says that "the corresponding area enclosed between the curve and the -axis is rotated about the -axis" and this sounds to me like the area bounded by the curve and the -axis between and is the volume being sought, rather than rotating the region bounded by the curve, the line , and the coordinates axes. If it is the region whose area was found in part (a) that is rotated, then the first answer is sought... if the region rotated is the different (and smaller) region bounded the curve, the -axis, and , then the question is easier as the subtraction from the cylinder is not needed, and the latter answer above applies.

Hmmm yes it all comes down to what the question writer meant by “corresponding” area. I’d just assumed it meant the same area from part a) but perhaps that’s not what they meant. Bloody confusing to use the word “corresponding” though if they meant just the area between the curve and the y-axis.

 

[CM_Tutor]

Hmmm yes it all comes down to what the question writer meant by “corresponding” area. I’d just assumed it meant the same area from part a) but perhaps that’s not what they meant. Bloody confusing to use the word “corresponding” though if they meant just the area between the curve and the y-axis.

I agree that it is ambiguous. The word "corresponding" is a strange choice if they meant the same region as in (a), though. That's why I asked what the given answer was. The source of the question would be useful to know, too - there are some texts that are better written than others, as I am sure you are aware.

@Cherrybomb56, can you tell us where the question is from, or what answer was given? Thanks.

 

[Cherrybomb56]

I agree that it is ambiguous. The word "corresponding" is a strange choice if they meant the same region as in (a), though. That's why I asked what the given answer was. The source of the question would be useful to know, too - there are some texts that are better written than others, as I am sure you are aware.

@Cherrybomb56, can you tell us where the question is from, or what answer was given? Thanks.

The question is from infocus. This was the answer.

 

[CM_Tutor]

The question is from infocus. This was the answer.

Thanks for letting us know, Cherrybomb.

From that answer, it seems clear that the question meant the region enclosed by the line , the -axis, and the curve in the first quadrant, is what is rotated about the -axis.

 

[Cherrybomb56]

Thanks for letting us know, Cherrybomb.

From that answer, it seems clear that the question meant the region enclosed by the line , the -axis, and the curve in the first quadrant, is what is rotated about the -axis.

So we wouldn't use the method in the video?

 

[CM_Tutor]

The video describes two steps, the first rotating the line to get the black slices of the cylinder and the second the rotation of the curve itself to get red slices. The problem is actually asking just the red slices solid. The video is correct if the region being rotated had been the region in part (a)... but it wasn't.