Solutions - 0100-01

(1) Integral formula:

$$F={\int }_a^b\rho gh(y)w(y)dy$$

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(2) Integrand components:

Width function:

$$w(y)=0+\frac{1-0}{2}(y-3)\gg \gg \frac{1}{2}y-\frac{3}{2}$$

Depth function: $h(y)=y$

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(3) Integrate:

$$\begin{array}{c}F={\int }_3^5(900\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)(9.8\mathrm{m}/{\mathrm{s}}^2)y(\frac{1}{2}y-\frac{3}{2})dy\\ \\ \gg \gg 8820{\int }_3^5\frac{1}{2}{y}^2-\frac{3}{2}ydy\gg \gg 8820{(\frac{1}{6}{y}^3-\frac{3}{4}{y}^2)|}_3^5\\ \\ \gg \gg 38220\mathrm{k}\mathrm{g}\cdot \mathrm{m}/{\mathrm{s}}^2\end{array}$$