Finding the Centroid of a Homogeneous Lamina

Find the centroid of the homogeneous lamina bounded by the graph of \( f(x)=x^{2},\) the \(x\)-axis, and the line \(x=1\).

Solution The lamina is shown in Figure 71.

The area \(A\) of the lamina is \begin{equation*} A=\int_{0}^{1}x\,^{2}{\it dx}=\left[ \dfrac{x^{3}}{3}\right] _{0}^{1}=\dfrac{1}{3} \end{equation*}

Using formulas (2), the centroid of the lamina is \begin{equation*} \bar{x} = \dfrac{1}{A}\int_{0}^{1}xf(x)~{\it dx}=\dfrac{1}{\dfrac{1}{3}} \displaystyle \int_{0}^{1}~x\cdot x^{2}\,{\it dx}=3\displaystyle \int_{0}^{1}x^{3}\,{\it dx}=3\left[ \dfrac{x^{4}}{4}\right] _{0}^{1}=\dfrac{3}{4} \\[4pt] \bar{y} = \dfrac{1}{2A}\displaystyle \int_{0}^{1}[f(x)] ^{2}\,{\it dx}=\dfrac{1}{\dfrac{2}{3}}\displaystyle \int_{0}^{1}( x^{2}) ^{2}\,{\it dx}=\dfrac{3}{2} \displaystyle \int_{0}^{1}x^{4}\,{\it dx}=\dfrac{3}{2}\left[ \dfrac{x^{5}}{5}\right] _{0}^{1}=\dfrac{3}{10} \end{equation*}

The centroid of the lamina is \(\left( \dfrac{3}{4},\dfrac{3}{10}\right) .\)