James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

570 Chapter 8 Further Applications of Integration
Therefore, from Definition 1, the consumer surplus is
y 500
0
Blood Flow
fpsxd 2 Pg dx − y 500
s1200 2 0.2x 2 0.0001x 2 2 1075d dx
0
− y 500
s125 2 0.2x 2 0.0001x 2 d dx
0
− 125x 2 0.1x 2 2 s0.0001dS x 3
3DG0
− s125ds500d 2 s0.1ds500d 2 2 s0.0001ds500d3
3
In Example 3.7.7 we discussed the law of laminar flow:
− $33,333.33 n
vsrd −
P
4l sR2 2 r 2 d
which gives the velocity v of blood that flows along a blood vessel with radius R and
length l at a distance r from the central axis, where P is the pressure difference between
the ends of the vessel and is the viscosity of the blood. Now, in order to compute the
rate of blood flow, or flux (volume per unit time), we consider smaller, equally spaced
radii r 1 , r 2 , . . . . The approximate area of the ring (or washer) with inner radius r i21 and
outer radius r i is
2r i Dr where Dr − r i 2 r i21
500
r i
Îr
(See Figure 4.) If Dr is small, then the velocity is almost constant throughout this ring
and can be approximated by vsr i d. Thus the volume of blood per unit time that flows
across the ring is approximately
s2r i Drd vsr i d − 2r i vsr i d Dr
FIGURE 4
FIGURE 5
and the total volume of blood that flows across a cross-section per unit time is about
o n
2r i vsr i d Dr
i−1
This approximation is illustrated in Figure 5. Notice that the velocity (and hence the
volume per unit time) increases toward the center of the blood vessel. The approximation
gets better as n increases. When we take the limit we get the exact value of the flux (or
discharge), which is the volume of blood that passes a cross-section per unit time:
F − lim
nl ` on 2r i vsr i d Dr − y R
i−1
0
− y R P
2r
0 4l sR2 2 r 2 d dr
2r vsrd dr
− P
2l yR 0
− P
2l
F R4
2 2 R4
sR 2 r 2 r 3 d dr − P
FR r 2
2
2l 2 2 r 4
−
4G PR4
8l
r−R
4Gr−0
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