Chapter 9, Problem 7P

A triangular area is bounded by a straight line in the x -y plane, as showing Fig. P9.7(a).

(a) Find the equation of the line $y\left(x\right)$.

(b) Find the area A by integration,

$A={\displaystyle \underset{0}{\overset{b}{\int }}y\left(x\right)dx}$ (c) Find the centroid G by integration with vertical rectangles, as shown in Fig. P9.7(b); in other words, find

$\bar{x}=\frac{{\displaystyle \int xdA}}{A}=\frac{{\displaystyle \int xy\left(x\right)dx}}{A}$ and

$\bar{y}=\frac{{\displaystyle \int \frac{y}{2}dA}}{A}=\frac{\frac{1}{2}{\displaystyle \int {\left(y\left(x\right)\right)}^{2}dx}}{A}$.

(d) No solve for x as a function of y, and recalculate the y -coordinate of the centroid G by Integration with

horizontal rectangles, as shown in Fig. P9.7(c). i.e., $\bar{y}=\frac{{\displaystyle \int ydA}}{A}=\frac{{\displaystyle \int yx\left(y\right)dy}}{A}$.