Solutions - 0100-02

(1) Integral formula:

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

---

(2) Integrand components:

$$w(y)=2\sqrt{y},h(y)=1-y$$

So we have:

$$F=\rho g{\int }_0^{1}2\sqrt{y}(1-y)dy$$

---

(3) Integrate:

$$\begin{array}{c}\gg \gg \rho g{\int }_0^{1}2y^{1/2}-2y^{3/2}dy\\ \\ \gg \gg {\rho g(\frac{4}{3}{y}^{3/2}-\frac{4}{5}{y}^{5/2})|}_0^1\\ \\ \gg \gg \rho g\frac{8}{15}\gg \gg 5227\mathrm{k}\mathrm{g}\mathrm{m}/{\mathrm{s}}^2\end{array}$$

(Assuming $\rho =1000$ and $g=9.8$.)