I have two questions:

For the first question, why would you not divide the final answer by 2? Because the volume of revolution shown by Region R would be the volume of the large sphere subtracted by the volume of the smaller sphere, divided by 2.

For the second question (about the glass bottle), why wouldn't you find the volumes of each of the individual shapes I have shown in the image? The mark scheme does include these three shapes, but for the cylinders the top cylinder has a height of 10 cm and the second cylinder has a height of 14 cm. However, when the region I labelled 1 is revolved about the y-axis, I thought it would not include the region I scribbled in purple. Therefore, I thought the height of the top cylinder would have to be 14 cm to encompass this.

For the first question, why would you not divide the final answer by 2? Because the volume of revolution shown by Region R would be the volume of the large sphere subtracted by the volume of the smaller sphere, divided by 2.

For the second question (about the glass bottle), why wouldn't you find the volumes of each of the individual shapes I have shown in the image? The mark scheme does include these three shapes, but for the cylinders the top cylinder has a height of 10 cm and the second cylinder has a height of 14 cm. However, when the region I labelled 1 is revolved about the y-axis, I thought it would not include the region I scribbled in purple. Therefore, I thought the height of the top cylinder would have to be 14 cm to encompass this.

(edited 4 months ago)

Original post by twisterblade596

I have two questions:

For the first question, why would you not divide the final answer by 2? Because the volume of revolution shown by Region R would be the volume of the large sphere subtracted by the volume of the smaller sphere, divided by 2.

For the first question, why would you not divide the final answer by 2? Because the volume of revolution shown by Region R would be the volume of the large sphere subtracted by the volume of the smaller sphere, divided by 2.

R is rotated about the x-axis, not the y-axis.

Original post by twisterblade596

I have two questions:

For the second question (about the glass bottle), why wouldn't you find the volumes of each of the individual shapes I have shown in the image? The mark scheme does include these three shapes, but for the cylinders the top cylinder has a height of 10 cm and the second cylinder has a height of 14 cm. However, when the region I labelled 1 is revolved about the y-axis, I thought it would not include the region I scribbled in purple. Therefore, I thought the height of the top cylinder would have to be 14 cm to encompass this.

For the second question (about the glass bottle), why wouldn't you find the volumes of each of the individual shapes I have shown in the image? The mark scheme does include these three shapes, but for the cylinders the top cylinder has a height of 10 cm and the second cylinder has a height of 14 cm. However, when the region I labelled 1 is revolved about the y-axis, I thought it would not include the region I scribbled in purple. Therefore, I thought the height of the top cylinder would have to be 14 cm to encompass this.

The expression for "radius" in relation to the curve GF does not exclude the portion between the y-axis and the line x = 1.

Original post by old_engineer

R is rotated about the x-axis, not the y-axis.

The expression for "radius" in relation to the curve GF does not exclude the portion between the y-axis and the line x = 1.

The expression for "radius" in relation to the curve GF does not exclude the portion between the y-axis and the line x = 1.

Thank you for the first one. I never read the question properly :\.

For the second question, why does the curve not exclude this region?

Original post by TwisterBlade596

Thank you for the first one. I never read the question properly :\.

For the second question, why does the curve not exclude this region?

For the second question, why does the curve not exclude this region?

Well, it does depend on how you're calculating volumes, but if you are integrating (pi)x^2 with respect to y (between y = 14 and y = 18), then the expression for x^2 is effectively the squared distance from the y-axis to the curve GF. So the bit between x = 0 and x = 1 isn't excluded.

There is another method that involves calculating the volume by considering the bottle as a set of infinitely thin nested cylinders, and in that case integrating with respect to x, but I'll leave that on one side unless that is what you were trying to do.

Original post by old_engineer

Well, it does depend on how you're calculating volumes, but if you are integrating (pi)x^2 with respect to y (between y = 14 and y = 18), then the expression for x^2 is effectively the squared distance from the y-axis to the curve GF. So the bit between x = 0 and x = 1 isn't excluded.

There is another method that involves calculating the volume by considering the bottle as a set of infinitely thin nested cylinders, and in that case integrating with respect to x, but I'll leave that on one side unless that is what you were trying to do.

There is another method that involves calculating the volume by considering the bottle as a set of infinitely thin nested cylinders, and in that case integrating with respect to x, but I'll leave that on one side unless that is what you were trying to do.

So is that a general rule? That a curve does not have to touch the axes at which it is being revolved about to create a solid which includes the region between the end of the curve and that particular axes (if that makes sense). I was originally rotating about the y-axis, but say the curve is rotated about the x-axis, would the solid generated include the region inside the dotted lines and the curve, or just the curve?

Original post by TwisterBlade596

So is that a general rule? That a curve does not have to touch the axes at which it is being revolved about to create a solid which includes the region between the end of the curve and that particular axes (if that makes sense). I was originally rotating about the y-axis, but say the curve is rotated about the x-axis, would the solid generated include the region inside the dotted lines and the curve, or just the curve?

Yes to both questions. Remember that what you're doing when calculating volumes of revolution in this way is effectively to sum the volumes of a series of very thin discs (horizontal discs with their centres on the y axis in this case).

Original post by old_engineer

Yes to both questions. Remember that what you're doing when calculating volumes of revolution in this way is effectively to sum the volumes of a series of very thin discs (horizontal discs with their centres on the y axis in this case).

I think I get it now. Thank you so much for your help!

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