<s>[INST] Volume made be a Revolving Pentagon I was looking to the volume created if I revolve the area of the pentagon having vertices
(1,0), (2,2), (0,4), (-2, 2), (-1,0) about the x-axis.
My work:
I used the Second theorem of Pappus, which states that the volume created is equal to the product
of circumference of the circle described by the centroid and the area. In short: $V = (2\pi \bar y)( A)$
I need to get the centroid of the pentagon.
$$\bar x = \frac{1+2+0+(-2)+(-1)}{5} = 0$$
$$\bar y = \frac{0+2+4+2+0}{5} = \frac{8}{5}$$
The area of the polygon laid out in Cartesian Coordinates is:
$$A = \frac{1}{2} \left[ \begin{matrix}
        x_1 & x_2 & x_3 & ... & x_n & x_1\\
        y_1 & y_2 & y_3 & ... & y_n & y_1\\
        \end{matrix} \right] $$
Turns out, the area described by the pentagon above is 10 square units.
We can now get the volume of solid of revolution of pentagon:
$$V = 2 \pi \bar y A$$
$$V = 2 \pi \left(\frac{8}{5}\right) (10)$$
We now get $32 \pi$ cubic units. 
But my book said the volume is $\frac{104}{3} \pi$
Where did I go wrong?
 [/INST] It's
$$2\left(\frac{4\cdot\pi\cdot4^2}{3}-\frac{1\cdot\pi\cdot2^2}{3}-\frac{2\cdot\pi\cdot2^2}{3}\right)=\frac{104\pi}{3}$$
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