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Section 8.3 : Center Of Mass

Find the center of mass for each of the following regions.

1. The region bounded by \(y = {x^3}\) , \(x = - 2\) and the \(x\)-axis.
2. The triangle with vertices \((-2, -2)\), \((4, -2)\) and \((4,4)\).
3. The region bounded by \(y = {\left( {x - 2} \right)^2}\) and \(y = 4\).
4. The region bounded by \(y = \cos \left( x \right)\) and the x-axis between \(\displaystyle - \frac{\pi }{2} \le x \le \frac{\pi }{2}\).
5. The region bounded by \(y = {x^2}\) and \(y = 6 - x\).
6. The region bounded by \(y = {{\bf{e}}^{2x}}\) and the x-axis between \( - 1 \le x \le 1\).
7. The region bounded by \(y = {{\bf{e}}^{2x}}\) and \(y = - \cos \left( {\pi x} \right)\) between \(\displaystyle - \frac{1}{2} \le x \le \frac{1}{2}\).