Question

The velocity \(v\) of the flow of blood at a distance \(r\) from the center of an artery of radius \(R\) can be modeled by \(v=k\left(R^{2}-r^{2}\right), \quad k>0\) where \(k\) is a constant. Find the average velocity along a radius of the artery. (Use 0 and \(R\) as the limits of integration.)

Short answer

The average velocity of the flow of blood at a distance from the center of an artery of radius \(R\) is given by \(\frac{2kR^2}{3}\)

Step-by-step solution

Set up the average value formula

The average velocity \(v_{avg}\) of blood flow along the radius of the artery is given by the formula \[v_{avg}=\frac{1}{R-0}\int_{0}^{R} v \, dr= \frac{1}{R}\int_{0}^{R} k \left(R^{2}-r^{2}\right) \, dr\].

Simplify and evaluate the integral

Next, we simplify the integral and then evaluate it: \[v_{avg}=\frac{k}{R} \int_{0}^{R} \left(R^{2}-r^{2}\right) \, dr = \frac{k}{R} \left. \left[\frac{R^2 r - r^3 /3}{1}\right]\right|_0^R.\] To get the result, we substitute \(R\) and \(0\) into this expression.

Calculate average velocity

Substituting the limits of integration yields: \[v_{avg} = \frac{k}{R} \left[(R^3 - R^3 /3) - 0\right] = \frac{k}{R} \frac{2R^3}{3} = \frac{2kR^2}{3}\].

Key concepts

Integration in Calculus

Integration in calculus is a fundamental process used to determine the accumulation of quantities, such as area under a curve, total distance traveled, or, in the context of our exercise, the average velocity across a distance. It is mathematically represented as an integral.

In the given blood flow problem, we use definite integration to find the average velocity of blood along a radius of the artery. Definite integration calculates the net value of a continuous function within specified limits, which in this case are 0 and the radius, R. By integrating the velocity function and dividing by the interval length (which is R), we obtain the average velocity, v_avg.

This process is fundamental in various applications of calculus, proving invaluable for solving real-world problems that require summing up small and infinite quantities to find a total effect, a procedure that is often used in physics, engineering, economics, and numerous other fields.

Applied Mathematics

Applied mathematics involves the use of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In our exercise, applied mathematics is used to model physical processes—in this case, the blood flow dynamics within an artery.

Mathematical modeling involves creating equations to represent real-world phenomena, thus enabling us to perform calculations that provide insights into the workings of real systems. With the use of the velocity function v=k(R²-r²), where k is a constant, we can model the velocity of blood at any point r within the artery.

Moreover, applied mathematics not only helps in modeling but also in predicting outcomes, optimizing systems, and solving complex problems that might be infeasible or impractical through empirical methods alone. For example, the use of integration to calculate the average velocity gives a precise value that is necessary for medical diagnostics and treatment.

Flow Dynamics

Flow dynamics, a subfield of fluid mechanics, is concerned with the study of how fluids move and interact with their environment. It applies principles of velocity, pressure, density, and temperature to analyze fluid flow.

In our context, flow dynamics pertains to the movement of blood within the circulatory system, specifically through an artery. The modeling equation v=k(R²-r²) reflects a simplified version of the intricate flow dynamics that occur in blood circulation. Here, velocity decreases from the maximum at the center (when r=0) to zero at the artery wall (when r=R), which is characteristic of a laminar flow profile under a parabolic velocity distribution.

Understanding these dynamics using calculus and mathematical modeling is vital in numerous applications, such as designing medical equipment, understanding cardiovascular conditions, and creating simulations to predict how alterations in the system (like artery blockages) can affect blood flow and overall health.