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Calculus/Simpsons Rule applied to biology

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Expert: - 2/13/2009


Question
After an 8-mg injection of dye, the readings of dye concentration, in mg/L, at two-second intervals are shown in the table. Use Simpson's rule to estimate the cardiac output.

If t=0 then c(t)=0. From here on in expressed as (t,c(t)):
(0,0)
(2,2.4)
(4,5.1)
(6,7.1)
(8,7.6)
(10,5.4)
(12,3.9)
(14,2.3)
(16,1.6)
(18,0.7)
(20,0)

Answer
Simpson's rule states that on any interval (a,b), the integral
of f(x) on (a,b) is (b-a)(f(a) + 4f((a+b)/2) + f(b))/6.

To do this, take (a,b) as (0,4), (4,8), (8,12), (12,16), and (16,20).
In doing so, (a+b)/2 will be 2, 6, 10, 14, and 18.

The evaluation would be

[f(0) + 4f(2) + 2f(4) + 4f(6) + 2f(8) + 4f(10) +
2f(12) + 4f(14) + 2f(16) + 4f(18) + f(20)]/(6*4).

The 6 is from the Simpson's Rule (since there are 6 points on top)
and the 4 is do to the interval width.  Note the f(4), f(8), f(12),
and f(16) occur on the left of one interval and the right side of the other interval.  That is why they have a 2 in front of them.  

Calculus

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