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Bread18

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Problem solved

I'm not sure how easy this will be to understand without a diagram, but I don't know how to upload one :(

Let a and b be constants, with a > b > 0.A torus is formed by rotating the

circle (x - a)^2 + y^2 = b^2 about the y-axis.

The cross-section at y = h, where –b ≤ h ≤ b, is an annulus. The annulus has

inner radius x1 and outer radius x2 where x1 and x2 are the roots of

(x - a)^2 = b^2 + y^2

(i) Find x1 and x2 in terms of h.

(ii) Find the area of the cross-section at height h, in terms of h.

(iii) Find the volume of the torus.

I think they're all up there.

I did part (i) and got x1 = a - [itex]\sqrt{b^2 - h^2}[/itex]

and x2 = a + [itex]\sqrt{b^2 - h^2}[/itex], which is correct according to the answers.

I did part (ii) and got A = [itex]\pi[/itex]a[itex]\sqrt{b^2 - h^2}[/itex], which again is correct.

But for part 3 I thought V = [itex]\pi[/itex]a[itex]\int^{b}_{-b}[itex]\sqrt{b^2 - h^2}dh[/itex] (sorry I don't know how to get rid of that itex in the integral)

But in the answers it is V = [itex]\pi[/itex]a[itex]\int^{a}_{-a}[itex]\sqrt{b^2 - h^2}dh[/itex], and I am not sure what it's between -a and a. The answers are from a different source then the question so I'm not sure if they have made a mistake or not.

Any help will be much appreciated, thanks.

I'm not sure how easy this will be to understand without a diagram, but I don't know how to upload one :(

## Homework Statement

Let a and b be constants, with a > b > 0.A torus is formed by rotating the

circle (x - a)^2 + y^2 = b^2 about the y-axis.

The cross-section at y = h, where –b ≤ h ≤ b, is an annulus. The annulus has

inner radius x1 and outer radius x2 where x1 and x2 are the roots of

(x - a)^2 = b^2 + y^2

(i) Find x1 and x2 in terms of h.

(ii) Find the area of the cross-section at height h, in terms of h.

(iii) Find the volume of the torus.

## Homework Equations

I think they're all up there.

## The Attempt at a Solution

I did part (i) and got x1 = a - [itex]\sqrt{b^2 - h^2}[/itex]

and x2 = a + [itex]\sqrt{b^2 - h^2}[/itex], which is correct according to the answers.

I did part (ii) and got A = [itex]\pi[/itex]a[itex]\sqrt{b^2 - h^2}[/itex], which again is correct.

But for part 3 I thought V = [itex]\pi[/itex]a[itex]\int^{b}_{-b}[itex]\sqrt{b^2 - h^2}dh[/itex] (sorry I don't know how to get rid of that itex in the integral)

But in the answers it is V = [itex]\pi[/itex]a[itex]\int^{a}_{-a}[itex]\sqrt{b^2 - h^2}dh[/itex], and I am not sure what it's between -a and a. The answers are from a different source then the question so I'm not sure if they have made a mistake or not.

Any help will be much appreciated, thanks.

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