[Maths HL] Ratios of Areas and Volumes

[Austin Glau]

For the Investigating Ratios of Areas and Volumes Portfolio, I got the Conjecture for ratios of areas A and B, and I got to 4. But it says

4. Are there general formulae for the ratios of the volumes of revolution generated by the regions A and B when they are each rotated about

(a) the x-axis?

(b) the y-axis?

State and prove your conjecture.

How should I set this up the proof?

[Marius]

Have you made a conjecture yet? I'm assuming you have, since you're asking about the proof. If you do, maybe you could tell us what it is, so we can help you better?

[Austin Glau]

Have you made a conjecture yet? I'm assuming you have, since you're asking about the proof. If you do, maybe you could tell us what it is, so we can help you better?

Yes, i did get a conjecture. It was, when y=x^n, the ratio of A and B was n:1

[Marius]

That's for part 3, right? So now you need to find the ratio between the volumes of revolution you get when you rotate the areas around the axes.

Umm... actually, I'm a bit unsure exactly what the problem is. Do you have problems making the conjecture, or proving it?

[Austin Glau]

Umm... actually, I'm a bit unsure exactly what the problem is. Do you have problems making the conjecture, or proving it?

I have problems making the conjecture. I got that Volume of A is 1145pi and volume of b is 310pi for the x-axis, which doesn't hold for my conjecture (B should be 310*n=310*3=930). I got a similarly odd answers for y-axis, 89pi and 119.2pi.

So my n:1 conjecture doesn't work.

I think my problem might be how I am getting the volume. I use

pi*integral(b,a)[f(x)^2-g(x)^2]

(the washer method or something like that)

Where f(x) is the top function,b^n, and g(x) is the function I am testing.

Is this method applicable for this kind of problem, because Im not sure i did it right.

Edited November 24, 2009 by 2401 I Hate Tangents

[Marius]

I think the answer is supposed to be something like 2n:1 or n/2:1. You could try taking the ratios between other volumes that you're doing. For example, if you let Ax be the volume of A around the x-axis, and similarly with Bx, Ay and By, you could try taking Ax:Bx, Ax:By and Bx:By to see if you get anything interesting.

Otherwise, I think the method you're using is correct, at least for finding Bx. It's much harder to find By and Ax.

[Austin Glau]

Thanks. I found out that I messed up in the area calculation, and I used the same conjecture for both axis.

[JWND]

Umm... actually, I'm a bit unsure exactly what the problem is. Do you have problems making the conjecture, or proving it?

I have problems making the conjecture. I got that Volume of A is 1145pi and volume of b is 310pi for the x-axis, which doesn't hold for my conjecture (B should be 310*n=310*3=930). I got a similarly odd answers for y-axis, 89pi and 119.2pi.

So my n:1 conjecture doesn't work.

I think my problem might be how I am getting the volume. I use

pi*integral(b,a)[f(x)^2-g(x)^2]

(the washer method or something like that)

Where f(x) is the top function,b^n, and g(x) is the function I am testing.

Is this method applicable for this kind of problem, because Im not sure i did it right.

the eqn should be pi * integral(b~a)[f(x)^2] or pi * integral(b~a)[f(y)^2

[Mr. Fish Sticks]

Does anybody have the program to graph the volumes of revolution generated by the regions A and B?

[Austin Glau]

@JWND

Thanks, I guess I missed that

@Mr. Fish Sticks

I used Winplot. It generates graphs like the following

Edited February 24, 2010 by 2401 I Hate Tangents