Answer
a) The total flow \( Q \) in the artery is given by the definte integral:
\[
Q = \int_{0}^{R} 2\pi k (R^2 - r^2) r \, dr
\]
b) The evaluated definite integral for the total flow \( Q \) in the artery is:
\[
Q = \frac{\pi k R^4}{2}
\]
Work Step by Step
a) To find the total flow of blood in the artery, we will integrate the flow contributions from each concentric layer over the entire radius of the artery. Given that the velocity of blood at a distance \( r \) from the center of the artery is \( v(r) = k(R^2 - r^2) \), and the differential area of a layer at radius \( r \) with thickness \( \delta r \) is \( dA = 2 \pi r \, dr \), we can set up the integral as follows:
1. Expression for the flow in a single layer:
The flow \( dQ \) in a thin layer of thickness \( dr \) at radius \( r \) is the product of the velocity and the differential area:
\[
dQ = v(r) \cdot dA = k(R^2 - r^2) \cdot 2\pi r \, dr
\]
2. Total flow \( Q \) in the artery:
To find the total flow, we need to integrate \( dQ \) from \( r = 0 \) (the center of the artery) to \( r = R \) (the outer wall of the artery):
\[
Q = \int_{0}^{R} dQ = \int_{0}^{R} k(R^2 - r^2) \cdot 2\pi r \, dr
\]
3. Simplifying the integrand:
Simplify the integrand \( k(R^2 - r^2) \cdot 2\pi r \):
\[
Q = \int_{0}^{R} 2\pi k (R^2 - r^2) r \, dr
\]
This definite integral gives the total flow of blood in the artery in terms of the numerical constant \( k \) and the radius \( R \) of the artery.
b) Let's evaluate the definite integral step-by-step.
Given:
\[
Q = \int_{0}^{R} k(R^2 - r^2) \cdot 2\pi r \, dr
\]
First, simplify the integrand:
\[
Q = \int_{0}^{R} 2\pi k (R^2 - r^2) r \, dr
\]
Expanding the integrand, we get:
\[
Q = 2\pi k \int_{0}^{R} (R^2 r - r^3) \, dr
\]
Now, we integrate term by term:
\[
Q = 2\pi k \left[ \int_{0}^{R} R^2 r \, dr - \int_{0}^{R} r^3 \, dr \right]
\]
Evaluate each integral separately:
1. For the first integral:
\[
\int_{0}^{R} R^2 r \, dr = R^2 \int_{0}^{R} r \, dr = R^2 \left[ \frac{r^2}{2} \right]_{0}^{R} = R^2 \left( \frac{R^2}{2} \right) = \frac{R^4}{2}
\]
2. For the second integral:
\[
\int_{0}^{R} r^3 \, dr = \left[ \frac{r^4}{4} \right]_{0}^{R} = \frac{R^4}{4}
\]
Substituting these results back into the expression for \( Q \):
\[
Q = 2\pi k \left( \frac{R^4}{2} - \frac{R^4}{4} \right)
\]
Simplifying the terms inside the parentheses:
\[
Q = 2\pi k \left( \frac{2R^4}{4} - \frac{R^4}{4} \right) = 2\pi k \left( \frac{R^4}{4} \right) = \frac{\pi k R^4}{2}
\]
Therefore, the evaluated definite integral for the total flow \( Q \) in the artery is:
\[
Q = \frac{\pi k R^4}{2}
\]
This is the final result for the total flow of blood in the artery in terms of the numerical constant \( k \) and the radius \( R \) of the artery.
