- Thread starter david123
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I suppose you are referring to "hints" provided for solution - or you are looking at the solution.

You are correct. You don't have to shift the ellipse. Have you tried to solve the problem without shifting?

If you do not shift - what answer do you get?

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And after shifting...

The volumes will be different. The question requires the volume of the second shape.

Don't worry, both of these shapes have been safely stashed away in area 51 .

NB: HallsofIvy's approach would probably work too.

I just spotted that I rotated the ellipse around x=5, not y=5. Off to the corner with me! (or perhaps to area 51?)The formula for "volume of revolution" usually considers a revolution around the x or y axis (in my experience). Here is what the ellipse equation looks like when revolved around the y axis...

View attachment 16608

And after shifting...

View attachment 16609

The volumes will be different. The question requires the volume of the second shape.

Don't worry, both of these shapes have been safely stashed away in area 51 .

However the pricipal of my posts should still hold even though the shapes will look different. If I get time later I'll post a corrected 3d view.

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I've always assumed you were **from** area 51!

And to calculate the volume of the above (left) shape, take the difference of the following two volumes:- LHS minus RHS.

These shapes correspond to the lower and upper half of the ellipse being revolved separately.

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Tires for your alien spaceship?See below for rotation around the x axis. The left shape has had the shift applied before rotation...

View attachment 16613

And to calculate the volume of the above (left) shape, take the difference of the following two volumes:- LHS minus RHS.

View attachment 16614

These shapes correspond to the lower and upper half of the ellipse being revolved separately.

and a cigar for when the calculation is complete!Tires for your alien spaceship?

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Approximately sketch the ellipse - the major axis of the ellipse is x-axis.

When you rotate the ellipse about y = 5, the "tire" above will be coming-out and going-in through z-direction. If you take a cross-section of the rotated-volume by the x-y plane - you will get two ellipses on the x-y plane. One with the major axis along y=0 (x-axis) and the other will be x-axis along y = 10. I think the shift you are talking about is at y=10. Then use washer method to calculate volume.

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In fact if you were given, for example, \(\displaystyle \dfrac{(x-2)^2}{4} + \dfrac{(y-3)^2}{9}=1\) and you want the volume you can rotate about y=3 and this will get you the correct result. However you can, if you like, move the ellipse so that its center goes from (2,3) to (0,0) by using the equation \(\displaystyle \dfrac{x^2}{4} + \dfrac{y^2}{9}=1\) and rotate about y=0. You can do this for any volume problem.

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Here is the difference. When you rotate a function (a 3-d figure) about y=5 you areI got confused because in calculus when you rotate a function around a line you don't usually move the function anywhere, it stays in one spot.

Now back to you original problem. Since you want the volume of the ellipse, NOT the volume of what you get when you revolve the ellipse about y=5, you can NOT revolve the original equation of the ellipse unless it happens to be centered along y=5.

So what do you do? You can recenter the ellipse so the y value of the center is at 5 and then compute your volume. But as I said in my last post you can now recenter this ellipse to the origin (or any point you like) and compute the volume.

Since one can move a figure to any location they want before rotating then this problem has got to be the winner of the most stupid problem I ever saw.

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But 51 is not a perfect 51. Ah I get it now, cubist is not perfect.I've always assumed you werefromarea 51!

Drat, this probably means that I'm also not a Mersenne primeBut 51 is not a perfect 51. Ah I get it now, cubist is not perfect.

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Of course I meant that 51 is not a perfect cube!But 51 is not a perfect 51. Ah I get it now, cubist is not perfect.

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Of course I meant that 51 is not a perfect cube!

Me and my brother used to have conversations about the numbers we liked best--I liked 12 because it can be divided by 1,2,3, and 4.

Then he said he liked 31. Just because I can't divide it.