Welcome back to Professor Baker’s Math Class! In Section 8-3, we are moving beyond pure geometry and stepping into the world of physics and engineering. Integration is not just about finding the area under a curve; it is a powerful tool used to calculate forces on dams, balance points of machinery, and volumes of complex 3D shapes. Let’s break down the two main applications covered in the attached class notes.

1. Hydrostatic Pressure and Force

Have you ever noticed that when you dive deep into a swimming pool, you feel more pressure in your ears? This is known as hydrostatic pressure. The deeper an object is submerged in a fluid, the greater the weight of the water above it.

For a horizontal plate, the pressure is constant. However, for a vertical plate (like a dam wall), the pressure increases as you go down. To find the total force, we must use calculus to sum up the forces acting on thin horizontal strips of the dam.

Key formulas to remember:

  • Pressure ($P$): Defined as force per unit area. At a specific depth $d$, the pressure is given by $$P = \rho g d$$ where $\rho$ is the fluid density and $g$ is the acceleration due to gravity.
  • Total Hydrostatic Force ($F$): By integrating the pressure over the area of the plate, we get: $$F = \int_a^b \rho g (\text{depth}) \cdot (\text{width}) \, dx$$

The notes include a detailed example of calculating the force on a trapezoidal dam. Pay close attention to how we set up the similar triangles to find the width of the dam at any given depth!

2. Moments and Centers of Mass

The second major topic is finding the point at which a thin plate (a lamina) balances perfectly. This point is called the center of mass (or centroid). We start with the Law of the Lever (think of a seesaw), where balance occurs when:

$$m_1 d_1 = m_2 d_2$$

In Calculus, we extend this to continuous shapes. To find the coordinates $(\bar{x}, \bar{y})$ of the centroid of a region bounded by a function $f(x)$, we calculate the moments about the axes and divide by the total area ($A$).

The Formulas:

  • Moment about the y-axis ($M_y$): Measures the tendency to rotate around the y-axis. $$M_y = \int_a^b x f(x) \, dx$$
  • Moment about the x-axis ($M_x$): Measures the tendency to rotate around the x-axis. $$M_x = \int_a^b \frac{1}{2} [f(x)]^2 \, dx$$
  • Coordinates of the Centroid: $$\bar{x} = \frac{1}{A} \int_a^b x f(x) \, dx, \quad \bar{y} = \frac{1}{A} \int_a^b \frac{1}{2} [f(x)]^2 \, dx$$

The Theorem of Pappus

Finally, the notes introduce a fascinating shortcut called the Theorem of Pappus. This theorem connects centroids to volumes of revolution (from Chapter 6). It states that the volume of a solid of revolution is the product of the area of the region ($A$) and the distance traveled by its centroid ($d$):

$$V = A \cdot d = A \cdot (2\pi \bar{r})$$

This is incredibly useful for finding the volume of shapes like a torus (a donut shape), as shown in Example 7 of the slides.

Make sure to review the step-by-step examples in the PDF notes attached below. Understanding how to set up these integrals is half the battle. Happy studying!