the center, the axis of $x$ horizontal and the axis of $y$ positive downward. The element of pressure is

$$2kyx\,dy$$

and the total pressure is

$$P\,=\,2k\!\int_{0}^{6}yx\,dy.$$

$x$ is expressed in terms of $y$ by means of the equation of the ellipse,

$$\frac{x^{2}}{64}+\frac{y^{2}}{36}=1.$$

Then

$$P\,=\,2k\,\frac{4}{3}\!\int_{0}^{6}\!y\sqrt{36\,-y^{2}}\,dy.$$

**Exercises**

**1.** Find the pressure on the vertical parabolic gate, Fig. 51: $(a)$ if the edge $AB$ lies in the surface of the water; $(b)$ if the edge $AB$ lies 5 feet below the surface.

**2.** Find the pressure on a vertical semicircular gate whose diameter, 10 feet long, lies in the surface of the water.

**73.** Arithmetic Mean.** The arithmetic mean, $A$, of a series of $n$ numbers, $a_{1}$, $a_{2}$, $a_{3}$, $\cdot\cdot\cdot$, $a_{n}$, is defined by the equation

$$nA\,=\,a_{1}+a_{2}+a_{3}+\cdot\cdot\cdot\cdot\cdot\cdot+a_{n}.$$

or

$$A\,=\,\frac{a_{1}+a_{2}+a_{3}+\cdot\cdot\cdot\cdot+a_{n}}{n}.$$

That is, $A$ is such a number that if each number in the sum