If velocity v of blood flow at distance r from the center of an artery of radius R is given by
v = k (R^2 - r^2), constant k > 0, find the average velocity v along the radius of an artery using 0 to R as the limits of integration
To find the average velocity along the radius of an artery, we need to calculate the average of the velocity function v over the interval [0, R].
The average velocity V_avg can be found using the following integral formula:
V_avg = (1 / (R - 0)) * ∫[0, R] v dr
In this case, the velocity function v is given by v = k (R^2 - r^2), where k is a constant and r represents the distance from the center of the artery.
Substituting the velocity function into the integral formula, we have:
V_avg = (1 / R) * ∫[0, R] k (R^2 - r^2) dr
Let's now solve this integral step by step.
Taking the constant k outside the integral, we get:
V_avg = k * (1 / R) * ∫[0, R] (R^2 - r^2) dr
Expanding the integrand, we have:
V_avg = k * (1 / R) * ∫[0, R] (R^2 - r^2) dr
= k * (1 / R) * [ ∫[0, R] R^2 dr - ∫[0, R] r^2 dr ]
Integrating each term separately, we have:
V_avg = k * (1 / R) * [ R^2 * r |_[0, R] - (1/3) * r^3 |_[0, R] ]
Applying the limits of integration, we get:
V_avg = k * (1 / R) * [ R^2 * R - (1/3) * R^3 - R^2 * 0 - (1/3) * 0^3 ]
= k * (1 / R) * [ R^3 - (1/3) * R^3 ]
= k * (1 / R) * [ (2/3) * R^3 ]
= (2/3) * k * R^2
Therefore, the average velocity V_avg along the radius of the artery is (2/3) * k * R^2.