I am having trouble understanding the concept. Usually when I calculate the center of mass of an object when given area and dimensions I'd multiply corresponding distances with areas etc then divide-by the total are (Given that we're using a 2D shape).
However when I come to this question for example:
I worked out the following: $(3*6^{2})-({\pi*3^{2}\over 2})(6-{4\over \pi})$ which is meant to give the center of mass, but I was under the assumption you must divide by the total area as per usual?
May someone explain to me why this isn't the case?
I can't tell how you're calculating it, but I find the CoM $y$-coord, $y_M$, as follows. Firstly, the area is easily calculated (without calculus):
\begin{eqnarray*} A &=& 6^2 - \dfrac{\pi \cdot 3^2}{2} = 36 - \dfrac{9\pi}{2}. \end{eqnarray*}
Now, assuming vertex $A$ is at $(-3,0)$ in the Cartesian plane,
\begin{eqnarray*} y_M &=& \dfrac{1}{A} \int_{-3}^{3} \dfrac{1}{2}(f(x)^2 - g(x)^2)\;dx \\ &=& \dfrac{1}{2A} \int_{-3}^{3} (6^2 - (9-x^2))\;dx \\ &=& \dfrac{1}{2A} \int_{-3}^{3} (x^2 +27)\;dx \\ &=& \dfrac{1}{2A} \bigg[\dfrac{x^3}{3} + 27x \bigg]_{-3}^{3} \\ &=& \dfrac{9+81}{A} \\ &=& \dfrac{20}{8-\pi} \\ && \\ \therefore\quad \text{Distance to CoM from $CD$ is } && 6 - \dfrac{20}{8-\pi}. \end{eqnarray*}