Welcome back to class! In this session, we moved away from abstract integration techniques and started looking at the powerful ways calculus describes the world around us. We covered Sections 8-3, 8-4, and 8-5, focusing on applications in economics, probability, and physics. It is fascinating to see how the same mathematical tool—the integral—can calculate money saved by shoppers, predict biological trends, and balance physical objects.

1. Economics: Consumer and Producer Surplus

We began by analyzing the interaction between supply and demand. In a perfect market, there is an equilibrium point $(X, P)$. However, the integral allows us to calculate the total benefit to the market.

  • Consumer Surplus: This represents the amount of money saved by consumers who were willing to pay more than the current market price. We calculate this by finding the area between the demand curve $p(x)$ and the horizontal line of the market price $P$.
    $$ \text{Consumer Surplus} = \int_{0}^{X} [p(x) - P] \, dx $$
    In class, we looked at a demand function for a product, $p = 1200 - 0.2x - 0.0001x^2$. At a sales level of 500 units, we found the surplus to be $33,333. We also applied this to an exponential demand function for microwave ovens.
  • Producer Surplus: On the flip side, this represents the benefit to manufacturers who were willing to sell for less than the market price. This is the area between the market price $P$ and the supply curve $p_s(x)$.
    $$ \text{Producer Surplus} = \int_{0}^{X} [P - p_s(x)] \, dx $$

2. Probability and The Normal Distribution

Next, we shifted gears to statistics and biology. We started with a "Net Change" application involving a mosquito population explosion modeled by $2200 + 10e^{0.8t}$, using the definite integral to find the total population increase between weeks 5 and 9.

However, the heavy hitter in this section was the Normal Distribution (the Bell Curve). The probability density function for a normal distribution is given by:

$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)} $$

Where $\mu$ is the mean and $\sigma$ is the standard deviation. We solved problems by setting up integrals to find the probability (area under the curve) for specific ranges:

  • IQ Scores: With $\mu=100$ and $\sigma=15$, we set up integrals to find the percentage of people with IQs between 85 and 115.
  • Pregnancy Lengths: Using $\mu=268$ days and $\sigma=15$ days.
  • Heights: We compared the distributions for adult men and women.

3. Physics: Center of Mass (Centroids)

Finally, we looked at finding the geometric center, or centroid, of a planar region. Think of this as the point where you could balance a cardboard cutout of the shape on the tip of a pencil.

For a region bounded by graphs, the coordinates of the centroid $(\bar{x}, \bar{y})$ are calculated using moments:

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

We practiced this by finding the centroid for regions bounded by curves like $y = \cos x$ and the area between lines and parabolas ($y=x$ and $y=x^2$).

Key Takeaway: Whether you are analyzing market efficiency, predicting population statistics, or engineering physical structures, the definite integral is your primary tool for accumulation and averaging. Keep practicing those setups!