Welcome back to Professor Baker's Math Class! In Section 8-4, we step away from abstract theory to see how Calculus II applies directly to the world around us. Whether you are interested in how markets determine value or how the human body functions, integration is the tool that makes these calculations possible.

1. Applications to Economics: Surplus

In economics, the interaction between consumers and producers creates a market equilibrium. We can use integrals to measure the total benefit gained by consumers and producers at this price point.

  • The Demand Function $p = d(x)$: Represents the price consumers are willing to pay for $x$ units.
  • The Supply Function $p = s(x)$: Represents the price at which producers are willing to supply $x$ units.
  • Equilibrium Point $(Q, P)$: The point where supply meets demand, found by solving $d(x) = s(x)$.

Consumer Surplus

This represents the amount of money saved by consumers who were willing to pay more than the actual market price. Geometrically, this is the area below the demand curve and above the price line $y = P$.

$$ \text{Consumer Surplus} = \int_0^Q [d(x) - P] \, dx $$

Producer Surplus

This represents the benefit to producers who were willing to sell at a price lower than the market price. Geometrically, this is the area below the price line $y = P$ and above the supply curve.

$$ \text{Producer Surplus} = \int_0^Q [P - s(x)] \, dx $$

2. Applications to Biology: Blood Flow

Calculus also helps us understand physiology. A classic application is modeling the flow of blood through a cylindrical artery (Poiseuille’s Law). Because of friction against the artery walls, blood flows faster in the center of the vessel and slower near the edges.

If we consider an artery with radius $R$ and length $L$, the velocity of the blood at a distance $r$ from the center is given by:

$$ v(r) = \frac{P}{4\eta L} (R^2 - r^2) $$

To find the total Flux (Flow Rate), we calculate the volume of blood passing a cross-section per unit of time. We visualize this by integrating concentric rings (shells) of blood flow from the center ($r=0$) to the wall ($r=R$).

$$ F = \int_0^R 2\pi r v(r) \, dx $$

By substituting $v(r)$ and evaluating this integral, we can determine the precise rate of blood flow, which is crucial for understanding cardiovascular health.

Key Takeaways

Remember, the power of integration lies in accumulation. Whether we are accumulating dollars saved in a market or volume of liquid moving through a tube, the process remains the same:

  1. Identify the function (Demand/Supply or Velocity).
  2. Set your limits of integration (usually 0 to Equilibrium Quantity or 0 to Radius).
  3. Set up the integral and evaluate.

Keep practicing these setups, and don't hesitate to reach out if you have questions about finding the equilibrium points!