Question A.1: For the triangular area of Fig. A.7a, determine (a) the firs......

a. First Moment Q_x. We choose to select as an element of area a horizontal strip with a length of u and thickness dy. Note that all of the points within the element are at the same distance y from the x axis (Fig. A.7b). From similar triangles,

\frac{u}{b}=\frac{h-y}{h} \quad u=b \frac{h-y}{h}

and

d A=u  d y=b \frac{h-y}{h} d y

The first moment of the area with respect to the x axis is

\begin{aligned}Q_x & =\int_A y  d A=\int_0^h y b \frac{h-y}{h} d y=\frac{b}{h} \int_0^h\left(h y-y^2\right) d y \\& =\frac{b}{h}\left[h \frac{y^2}{2}-\frac{y^3}{3}\right]_0^h Q_x=\frac{1}{6} b h^2\end{aligned}

b. Ordinate of Centroid. Recalling the first of Eqs. (A.4) Q_x=A \bar{y} \quad Q_y=A \bar{x} and observing that A=\frac{1}{2} b h,

\begin{gathered}Q_x=A \bar{y} \quad \frac{1}{6} b h^2=\left(\frac{1}{2} b h\right) \bar{y} \\\bar{y}=\frac{1}{3} h\end{gathered}