(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This project deals with custom made gold wedding bands. Its shape is obtained by revolving the region shown about a horizontal axis. The resulting band has Inner radius R, Minimum Thickness T, Width W.

The curved boundary of the region is an arc of a circle whose center lies on the axis of revolution. For a typical wedding band with given dimensions R, T, W, the jeweller must first calculate the volume of the desired wedding band to determine how much gold will be required.

Show that the volume V is given by the formula

[tex]\frac {\pi W}{6} (W^2 + 12RT + 6T^2)[/tex]

2. Relevant equations

[tex]x^2 + y^2 = R^2[/tex]

3. The attempt at a solution

First i tried to use the shell method and came up with this:

[tex]2 \cdot 2\pi \left(\int_{R}^{(R+T)}x\frac w2 \,dx + \int_{(R+T)}^{b}x\sqrt{b^2-x^2} \, dx \right)[/tex]

where b is the radius of the curved boundary

and that led to this:

[tex]\pi W(2RT+T^2) + \pi\frac 43(b^2 -(R+T)^2)[/tex]

But now i got stuck. I don't know what to do about the b.

I don't even know if i'm on the right track now.

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# Homework Help: Volume of revolution integral question

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