Chapter 8.4, Problem 22E

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# After a 5.5-mg injection of dye, the readings of dye concentration, in mg/L, at two-second intervals are as shown in the table. Use Simpson’s Rule to estimate the cardiac output.

To determine

To calculate: The cardiac output using Simpson’s rule.

Explanation

Given information:

The quantity of dye is 5.5â€‰mg.

Dye concentration is measured in mg/L.

Number of subintervals is 8.

Time is T=16â€‰sec.

Calculation:

The Simpsonâ€™s rule is shown below:

âˆ«abf(x)dxâ‰ˆSn=Î”x3[f(x0)+4f(x1)+2f(x2)+4f(x3)+...+2f(xnâˆ’2)+4f(xnâˆ’1)+f(xn)] (1)

The interval width Î”x=bâˆ’an (2)

Here, the value of the function f(x) at x=0â€‰toâ€‰n is f(x0)â€‰toâ€‰f(xn), the upper limit is b, the lower limit is a, and the number of interval is n.

Modify Equation (1) to calculate the dye concentration as shown below:

âˆ«abc(t)dtâ‰ˆSn=Î”t3[c(t0)+4c(t1)+2c(t2)+4c(t3)+...+2c(tnâˆ’2)+4c(tnâˆ’1)+c(tn)] (3)

Find the width of the interval:

Substitute 16 for b, 0 for a, and 8 for n in Equation (2).

Î”x=16âˆ’08=2

Thus the interval are, 0, 2 sec, 4 sec, 6 sec, 8 sec, 10 sec, 12 sec, 14 sec, and 16 sec.

Modify Equation (3) for n=8.

S8=Î”t3[c(t0)+4c(t1)+2c(t2)+4c(t3)+2c(t4)+4c(t5)+2c(t6)+4c(t7)+c(t8)] (4)

Substitute 0 for t0, 2 sec for t1, 4 sec for t2, 6 sec for t3, 8 sec for t4, 10 sec for t5, 12 sec for t6, 14 sec for t7, and 16 sec for t8 in Equation (4)

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