Problem 30 - Calculus Single Variable

Question

Use Simpson's Rule to estimate cardiac output based on the tabulated readings (with \(t\) in seconds and \(c(t)\) in \(\mathrm{mg} / \mathrm{L}\) ) taken after the injection of \(5 \mathrm{mg}\) of dye. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{c}(\boldsymbol{t}) & 0 & 3.8 & 6.8 & 8.6 & 9.7 & 10.2 & 9.4 & 8.2 & 6.1 & 3.1 & 0 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The estimated cardiac output is approximately 0.0752 liters per second.

Step 1: Understand Simpson's Rule

Simpson's Rule is a numerical method for approximating the integral of a function. It is given by:\[\int_a^b f(x) \, dx \approx \frac{\Delta x}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right)\]where \(\Delta x = \frac{b-a}{n}\), and \(n\) must be even.

Step 2: Identify Variables

In our problem, the variables are: \(a = 0\), \(b = 10\), and the function values are given in the table. The value of \(n\) should be 10 since the number of intervals (10 in this case) is even, which satisfies the requirement for Simpson's Rule.

Step 3: Calculate \(\Delta x\)

\(\Delta x\) is calculated based on the interval between time points:\[\Delta x = \frac{b-a}{n} = \frac{10 - 0}{10} = 1.0\]

Step 4: Apply Simpson's Rule

Substitute the values into Simpson's Rule:\[\int_0^{10} c(t) \, dt \approx \frac{1}{3} \left( 0 + 4(3.8) + 2(6.8) + 4(8.6) + 2(9.7) + 4(10.2) + 2(9.4) + 4(8.2) + 2(6.1) + 4(3.1) + 0 \right)\]Simplify it:

Step 5: Simplify Calculation

Continue simplifying:\[= \frac{1}{3} (0 + 15.2 + 13.6 + 34.4 + 19.4 + 40.8 + 18.8 + 32.8 + 12.2 + 12.4 + 0)\]\[= \frac{1}{3} (199.6) = 66.53\]

Step 6: Estimate Cardiac Output

Cardiac output is estimated using the formula:\[Cardiac \ Output = \frac{\text{Dose}}{\int_0^{10} c(t) \, dt}\]With the dose being \(5 \text{mg}\):\[\text{Cardiac Output} = \frac{5}{66.53} = 0.0752 \, \text{L/s}\]

Key Concepts

Numerical Integration

Numerical integration is a technique used to approximate the integral of a function when its analytical form is difficult or impossible to solve. It is especially useful in cases where data is available in a tabulated form, like a series of measurements over time. Instead of finding the exact area under a curve, numerical integration techniques—such as Simpson's Rule or the Trapezoidal Rule—estimate this area by summing up sections that approximate the curve.

This is useful in many real-world applications, like physics, engineering, and medical fields, where precise measurements can be very complex. By breaking down the integration process into simpler parts, these methods allow us to estimate integrals with more computable steps.

• Applications: Weather prediction, astrophysics, optimization problems.
• Methods: Simpson's Rule, Trapezoidal Rule, Midpoint Rule.

Cardiac Output Estimation

Estimating cardiac output is important in medical diagnostics as it gives insight into how much blood the heart pumps through the circulatory system per second. Physicians can assess heart function by measuring the concentration of a dye over time after it has been injected into the bloodstream.

Simpson's Rule is often used in this context. By integrating the concentration-time curve, one can deduce how much blood is being moved, which helps in calculating cardiac output.

• Procedure: A dye of known concentration is injected into the bloodstream.
• Measurement: The concentration of dye is recorded at different time intervals.
• Analysis: Use numerical integration to find the area under the concentration curve.

The formula for cardiac output is derived from these concentrations: \[Cardiac\ Output = \frac{\text{Dose injected}}{\int_{0}^{t} c(t) \, dt}\] Understanding this process allows healthcare professionals to evaluate cardiac health efficiently and non-invasively.

Trapezoidal Rule

The Trapezoidal Rule is another method of numerical integration, used to estimate the integral of a function over a given interval. It divides the area under a curve into trapezoids rather than the parabolic arcs that Simpson's Rule uses.With the Trapezoidal Rule, the formula is given by:\[\int_a^b f(x) \, dx \approx \frac{b-a}{2n} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right)\]This rule is simpler but less accurate than Simpson's Rule when fewer data points are used. However, for linear functions or cases where computation simplicity is a priority, it remains a reliable method.

• Advantages: Simplicity, better than midpoint for some functions.
• Ideal Use: When the number of function evaluations is a limiting factor.

Both Simpson’s Rule and the Trapezoidal Rule are frequently applied to different situations, depending on the precision needed and the nature of the data available. Understanding how to use these methods is essential for applications ranging from engineering problems to medical diagnostics.