Important Syllabus Update

Hello class! Before we dive into the math, please note a slight change to the syllabus. We are going to cover Chapter 8 instead of Chapter 6. Please don't worry—the format of the take-home test will remain exactly the same. This shift allows us to focus on some truly fascinating applications of the integration techniques you have been mastering.

Section 8-3: Applications to Physics and Engineering

In this section, we move from abstract integration to concrete physical reality. We will apply calculus to solve problems related to fluid pressure and the balance of physical objects. Here are the key concepts from the lecture and notes.

1. Hydrostatic Pressure and Force

Have you ever wondered about the force exerted by water against a dam or the viewing window of an aquarium? In physics, pressure ($P$) is defined as force per unit area. For fluids, pressure increases with depth according to the formula:

$$P = \rho g d$$

Where:

  • $\rho$ (rho) is the density of the fluid (e.g., $1000 \, kg/m^3$ for water).
  • $g$ is the acceleration due to gravity ($9.8 \, m/s^2$).
  • $d$ is the depth below the surface.

To find the Total Hydrostatic Force ($F$) against a vertical plate, we cannot simply multiply pressure by area because pressure changes as we go deeper. Instead, we slice the plate into horizontal strips and integrate:

$$F = \int_{a}^{b} \rho g \cdot (\text{depth}) \cdot (\text{width}) \, dy$$

2. Moments and Centers of Mass

The second major application in this section is finding the Center of Mass (often called the centroid for 2D geometric shapes). Think of this as the perfect balancing point of a thin plate (lamina).

To find this point $(\bar{x}, \bar{y})$, we calculate the moments about the x and y axes. The moment measures the tendency of the system to rotate.

  • Moment about the y-axis ($M_y$): Measures the distribution of mass relative to the y-axis.
  • Moment about the x-axis ($M_x$): Measures the distribution of mass relative to the x-axis.

The coordinates of the centroid are given by:

$$\bar{x} = \frac{M_y}{m} = \frac{1}{A} \int_{a}^{b} x [f(x) - g(x)] \, dx$$ $$\bar{y} = \frac{M_x}{m} = \frac{1}{A} \int_{a}^{b} \frac{1}{2} \{[f(x)]^2 - [g(x)]^2\} \, dx$$

(Note: In geometric problems where density is constant, the mass $m$ is replaced by the Area $A$).

Tips for Success

These problems rely heavily on setting up the integral correctly based on the geometry of the object. Sketching the shape and identifying your "sample slice" (the $dy$ or $dx$ strip) is crucial. Once the integral is set up, the evaluation is standard calculus!

Be sure to review the attached Section 8-3 Notes for specific examples and watch the video lecture where I walk through the derivation of these formulas step-by-step. Keep up the great work!