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(a). Write down a formula for the volume V of the solid obtained by revolving the region R about the y-axis. I think: $\displaystyle\int_{x=a}^{x=b}[f(x) - g(x)]*(2\pi(x))dx$

(b). Write down the formula for the moment M_y of R about the y-axis. I think: $M_y = \frac{\displaystyle\int_{x=a}^{x=b}[f(x) - g(x)]xdx}{\displaystyle\int_{x=a}^{x=b}[f(x) - g(x)]dx}$

c). Use the formula in (a) and (b) to derive the formula V = 2π(xbar)A. (A result of Pappus and Gulidin)?

Thank you

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    Do you know the theorem of pappus? from there it's fairly straightforward – 2017-02-14
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    Can you explain what I must do here. I still feel lost. Thank you – 2017-02-14

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The theorem of Pappus states that the volume of a solid of revolution $V=2\pi r A$ where A is the area of cross section and r is the distance from the center to the axis of revolution. So if I am revolving about the y axis then the distance from the center to the axis is just going to be the x coordinate of the center of mass otherwise known as $\bar{x}$ therefore our volume (using the variables you are using) is $2\pi \bar{x} M$