Answer to Question #260478 in Calculus for Artika

Question

. It is well known that the rate of flow can be found by measuring the volume of blood that flows past a point in a given time period. The volume V of blood flow through the blood vessel is 2pievr dr integral from R to 0 where v = K (R^2- r^2) is the velocity of the blood through a vessel. In the velocity of the blood through a vessel, K is a constant, the maximum velocity of the blood, R is a constant, the radius of the blood vessel and r is the distance of the particular corpuscle from the center of the blood vessel.

(i) If R = 0.30 cm and v= ( 0.30- 3.33 cm/s), find the volume.

(ii) Construct and develop a general formula for the volume V of the blood flow

Answer 1

(i)

$$V=\displaystyle\int_{0}^{0.30}2\pi r(0.30-0.33r^2)dr$$

$$=\pi\displaystyle\int_{0}^{0.30}(0.6r-0.66r^3)dr$$

$$=\pi[0.3r^2-0.165r^4]\begin{matrix} 0.30 \\ 0 \end{matrix}$$

$$=\pi((0.3(0.3)^2-0.165(0.3)^4)=0.0256635\pi(cm^3)$$

(ii)

$$V=\displaystyle\int_{0}^{R}2\pi Kr(R^2-r^2)dr$$

$$=2\pi K\displaystyle\int_{0}^{R}(R^2r-r^3)dr$$

$$=2\pi K\bigg[\dfrac{R^2r^2}{2}-\dfrac{R^4}{4}\bigg]\begin{matrix} R \\ 0 \end{matrix}$$

$$=\dfrac{\pi K R^4}{2}(cubic\ unts)$$