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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is

v ( r ) = P 4 η l ( R 2 r 2 )

where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood (see Example 2.7.7). Find the average velocity (with respect to r) over the interval 0 r R . Compare the average velocity with the maximum velocity.

To determine

To find:

(i) The average velocity over the interval 0rR

(ii) Compare the average velocity with the maximum velocity.

Explanation

1) Concept:

Use the mean value theorem for integral to find the average velocity over the interval 0rR

2) Mean Value Theorem for Integrals

If f is continuous on a,b then there exists a number c in a,b such that

fc=fave=1b-aabfx dx

that is,

abfx dx=fcb-a

3) Given:

The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is

vr=P4ηlR2-r2

where P is the pressure difference between the ends of the vessel and η is the viscosity of the blood.

4) Calculation:

(i)

The average velocity on 0,R is

Vave=1R-00Rvrdr

=1R0RP4ηlR2-r2 dr

=P4Rηl0RR2-r2 dr

=P4RηlR2r-r330R

=P4RηlR3-R330R

=P4Rηl2R33-0

=PR26ηl

Therefore,

Vave=PR26ηl

(ii)

vr=P4ηlR2-r2 is a polynomial of degree 2 in r

vr=PR24ηl-Pr24ηl

Differentiate vr with respect to r

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