Chapter 13.2, Problem 62E

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Velocity of blood In Problems 63 and 64, the velocity of blood through a vessel is given by v   = K ( R 2 − r 2 ) , where K is a constant, R is the (constant) radius of the vessel, and r is the distance of the particular corpuscle from the center of the vessel. The rate of flow can be found by measuring the volume of blood that flows past a point in a given time period. This volume, V, is given by V = ∫ 0 R υ ( 2 π r   d r ) Develop a general formula of V by evaluating V = ∫ 0 R υ ( 2 π r   d r ) using υ = K ( R 2 − r 2 ) .

To determine

To calculate: The general formula for volume of vessel if volume V is given by V=0Rv(2πr)dr, velocity of blood v=K(R2r2) cm/s.

Explanation

Given Information:

The volume V is given by V=0Rv(2πr)dr, v=K(R2r2) cm/s.

Formula used:

To calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

The value of the integral xndx=xn+1n+1.

Calculation:

Consider the volume V is given by V=0Rv(2πr)dr and v=K(R2r2) cm/s.

Evaluate the integral V=0Rv(2πr)dr.

Since, v=K(R2r2) .

Thus, the integral is

V=0R(K(R2r2))(2πr)dr=K0R(2πR2r2πr3)dr

Recall that to calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

Thus, evaluate the integral

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