Question

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.

Answer 1

$$A=-\displaystyle\int_{-2}^{0}x^3dx=-[\dfrac{x^4}{4}]\begin{matrix} 0 \\ -2 \end{matrix}=4$$

$$\bar{x}=\dfrac{1}{4}(-\displaystyle\int_{-2}^{0}x(x^3)dx)=-\dfrac{1}{4}[\dfrac{x^5}{5}]\begin{matrix} 0 \\ -2 \end{matrix}=-\dfrac{8}{5}$$

$$\bar{y}=\dfrac{1}{4}(-\dfrac{1}{2}\displaystyle\int_{-2}^{0}(x^3)^2dx)=-\dfrac{1}{8}[\dfrac{x^7}{7}]\begin{matrix} 0 \\ -2 \end{matrix}=-\dfrac{16}{7}$$

The centroid is $(-\dfrac{8}{5}, -\dfrac{16}{7}).$