Example 2.82. A vertical dam has the shape of an isosceles trapezoid with the upper base \( a = 60 \) units of length, the lower base \( b = 20 \) units of length, and height \( h = 10 \) units of length (Fig. 2.15). Calculate the force of water pressure on the dam.

![](https://cdn.mathpix.com/cropped/2024_05_22_0f84e17422d7c0df8d68g-104.jpg?height=320&width=453&top_left_y=147&top_left_x=642)

Fig. 2.15.
Solution. The area of the trapezoid element $L R T K$ is approximately replaced by the area of a rectangle with base $L R$ and height $d x$: $d S(x)=2\left(b+\frac{(a-b)(h-x)}{h}\right) d x$. According to Pascal's law, we obtain that the elementary force of pressure

$$
d F=\rho g x d S(x)=2 \rho g x\left(b+\frac{(a-b)(h-x)}{h}\right) d x
$$

$$
\text{Therefore } F=2 \int_{0}^{h} \rho g x\left(b+\frac{(a-b)(h-x)}{h}\right) d x=\rho g \frac{h^{2}}{3} \text {. }
$$