(adsbygoogle = window.adsbygoogle || []).push({}); There is a glass chamber that is to be filled with water. The chamber is divided into two parts, the outer being filled with water and the inner being empty. The chamber is such that a person can stand inside without getting wet.

It is modelled by the equations [tex]f(x) = \frac{243}{1820x^2} - \frac{243}{1820} \mbox{and} g(x) = \frac{2}{3}f(x) = \frac{486}{5460x^2} - \frac{486}{5460}[/tex]

The chamber has a height of 0.9m and a width of 1m, as shown in the attached image.

1. Find the total volume to fill the flask in cubic cm.

Alright, so what I did was rearrange each equation into terms of y as the solid is revolved around the y-axis.

Therefore:

[tex]f(y) = \sqrt{\frac{243}{1820\left(y + \frac{243}{1820}\right)}}[/tex]

[tex]g(y) = \sqrt{\frac{486}{5460 \left(y + \frac{486}{5460}\right)}}[/tex]

To find the volume, it is upper bound - lower bound, so:

[tex]V = \pi\int_{0}^{0.9} \left(\frac{243}{1820\left(y + \frac{243}{1820}\right)}\right) - \left(\frac{486}{5460\left(y + \frac{486}{5460}\right)}\right) dy[/tex]

It is at this point that I am stuck. I have tried to rearrange into one fraction, but then I am unable to integrate it. Maybe somebody could shed some light as to what I could do, or a simpler way of going about this? Thanks.

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# Volumes of Revolutions

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