Volumes question, volume of a torus

In summary, the problem involves finding the volume of a torus formed by rotating a circle around the y-axis. The cross-section at a given height is an annulus with inner and outer radii determined by the roots of a quadratic equation. The solution involves finding the roots and using them to calculate the area of the cross-section, which can then be integrated to find the volume.
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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 :(

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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  • #2
I just remembered an easy way to find the volume of a torus and my answers was right, don't need help anymore.
 

What is a torus?

A torus is a geometric shape that resembles a doughnut or a tire. It is a three-dimensional object with a circular cross-section and a hole in the middle.

How do you calculate the volume of a torus?

To calculate the volume of a torus, you can use the formula V = 2π²Rr², where R is the major radius (distance from the center of the torus to the center of the circular cross-section) and r is the minor radius (radius of the circular cross-section).

What is the difference between a torus and a cylinder?

A torus and a cylinder both have circular cross-sections, but a torus has a hole in the middle while a cylinder does not. Additionally, the cross-section of a torus is perpendicular to the axis, while the cross-section of a cylinder is parallel to the axis.

Can the volume of a torus be negative?

No, the volume of a torus cannot be negative as it is a measure of physical space and cannot have a negative value.

What are some real-life examples of a torus?

Some real-life examples of a torus include doughnuts, inner tubes, and the shape of a tire. It can also be seen in the design of certain architecture and sculptures.

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