Applications of Integration to Find Blood Flow

Blood flow in a blood vessel can be described using the law of laminar flow, where velocity varies with radial distance from the center.

The total blood flow can be approximated by dividing the cross-section into thin concentric rings with varying radii.

Each ring’s area is given by its circumference multiplied by its small thickness.

The flow through each ring can be approximated as the product of its area and the velocity at that radius.

Summing these flows for all rings gives an approximate total flow, called the Riemann sum.

As the number of rings approaches infinity, the Riemann sum approaches the exact values, represented by the definite integral from the center to the vessel wall.

The velocity function in the integral comes from the law of laminar flow. It states that the velocity at a distance r depends on the pressure difference, the fluid’s viscosity, and the vessel’s length.

Substituting the velocity expression into the integral and integrating over the vessel radius yields the exact blood flow.

This result, called Poiseuille’s Law, shows that flow scales with the fourth power of the vessel’s radius. This is why even a small narrowing of blood vessels sharply reduces blood flow.