Volume of revolution.

I'm stuck on this particular question

59.(a) set up an integral for the volume of a solid torus with radii r and R.

If you could see the diagram I'm looking at, its a donut, the area of a particular cut of the donut is

\pi r^2

. R is the going from point (0,0) and is touching the centre of the donut.

I'm thinking

A(x)=R-\pi r^{2}

and integrate this as a revolution

2\pi \int_{0}^{r} A(x) dr

But, then thats not with x. I don't know how to describe x. I'm thinking about just describing y, but then your rotating it around x not y to get the donut shape.

P.S. Tennis is not helping me concentrate.
P.P.S. But its really gay when things are rotated for example around say y=2 as that blood stupid. I think diagrams help.
P.P.P.S. My intuition about calculus has really improved, I particular like Riemann sum notation.

Reply 1

stevencarrwork

If R and r are both lengths, then R - pi*r^2 will have the wrong dimensions.

Isn't a torus just a cylinder folded so the two ends meet?

Reply 2

generalebriety

Consider a circle of radius r centred at (0, R). Rotate it about the x-axis.

Reply 3

Simplicity

Yeah, thats what I did.

(x-R)^2+y^2=r^2

Then, I rearrange to find y.

So we get

V=2 \pi \int_0^{r} (r^2 - x^2 +2xR -R^2)dx

But the book intergrates dy, and its different to mine.

I don't know I could rearrange for x as function of y, then maybe I don't know. Damn, you book.

Reply 4

generalebriety

Simplicity

Yeah, thats what I did.

(x-R)^2+y^2=r^2

Then, I rearrange to find y.

So we get

V=2 \pi \int_0^{r} (r^2 - x^2 +2xR +R^2

But the book intergrates dy, and its different to mine.

I don't know I could rearrange for x as function of y, then maybe I don't know. Damn, you book.

Well, what does the book do? It probably just flips the whole situation by putting a circle at (R,0) and rotating that about the y-axis.

Reply 5

Simplicity

Sorry edited it again.

Yeah, the book seems to do that. But, surely it doesn't matter and shouldn't matter for final anwsers. Mind is different to book, alot different.

Reply 6

generalebriety

Simplicity

Sorry edited it again.

Yeah, the book seems to do that. But, surely it doesn't matter and shouldn't matter for final anwsers. Mind is different to book, alot different.

Well, post your solution and post the book's solution and see where you differ. You'll probably find something like you've only rotated one half of the circle.