9 FURTHER APPLICATIONS OF INTEGRATION
9 FURTHER APPLICATIONS OF INTEGRATION
9 FURTHER APPLICATIONS OF INTEGRATION
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GROUP WORK, SECTION 9.4<br />
Homer’s Blood<br />
The volume of blood that passes through a cross-section of an artery (of radius R, with pressure difference<br />
P) per unit time is given by the formula<br />
F = πPR4<br />
8ηl<br />
(l is the length of the artery and η is viscosity of the blood — both are assumed to be constant).<br />
F is often called or defined as the flux. High blood pressure results from constriction of the arteries. As the<br />
radius of the arteries shrinks, the blood pressure needs to increase to maintain a constant flux.<br />
1. Why should we expect the flux to remain constant?<br />
2. Suppose P 0 and R 0 are the initial radius and the initial pressure. Then the flux is πP 0R0<br />
4 . If the pressure<br />
8ηl<br />
and radius at some other time are P and R, then the flux is πPR4 . Using the fact that flux must remain<br />
8ηl<br />
constant, determine an expression for P in terms of R, R 0 ,andP 0 .<br />
(<br />
3. As the years go by, the diameter of Homer’s arteries is given by R (t) = R 0 1 − t )<br />
,wheret is the<br />
100<br />
number of years from the present. What is Homer’s blood pressure in terms of his original blood pressure<br />
(a) 5 years from now?<br />
(b) 10 years from now?<br />
(c) 20 years from now?<br />
(d) 40 years from now?<br />
4. Homer’s doctor would like to perform angioplasty to enlarge the arteries when his blood pressure has<br />
doubled. When will this happen?<br />
510
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