Problem 28 - Calculus for Biology and Medicine

Question

Suppose that the concentration (measured in \(\mathrm{gm}^{-3}\) ) of nitrogen in the soil along a transect in moist tundra yields data points that follow a straight line with equation $$ y=673.8-34.7 x $$ for \(0 \leq x \leq 10\), where \(x\) is the distance to the beginning of the transect. What is the average concentration of nitrogen in the soil along this transect?

Step-by-Step Solution

Answer

The average concentration is \(500.3 \, \mathrm{gm}^{-3}\).

Step 1: Identify the Function

We start with the linear equation given for nitrogen concentration: \(y = 673.8 - 34.7x\). This describes how the concentration \(y\) changes with distance \(x\).

Step 2: Define the Integral for Average Value

To find the average concentration, we need to integrate the concentration function over the interval \([0, 10]\) and then divide by the length of the interval. The average value \(\bar{y}\) is given by \[\bar{y} = \frac{1}{b-a} \int_{a}^{b} y \, dx\], where \(a = 0\) and \(b = 10\).

Step 3: Set Up the Integral

Substitute the given function into the integral: \[\int_{0}^{10} (673.8 - 34.7x) \, dx\].

Step 4: Solve the Integral

Calculate the integral: \[\int 673.8 \, dx = 673.8x\] and \[\int -34.7x \, dx = -\frac{34.7}{2}x^2 = -17.35x^2\]. Combine these results to get \[673.8x - 17.35x^2 \bigg|_{0}^{10}\].

Step 5: Evaluate the Integral from 0 to 10

Plug in the limits into the integrated function: \[\left(673.8(10) - 17.35(10)^2\right) - (673.8(0) - 17.35(0)^2) = (6738 - 1735) - 0\].

Step 6: Compute the Result

Calculate the result from the subtraction to find the definite integral: \[6738 - 1735 = 5003\].

Step 7: Calculate the Average Concentration

Divide the integral result by the interval length: \[\bar{y} = \frac{5003}{10 - 0} = \frac{5003}{10} = 500.3\].

Key Concepts

Integration

Integration is a powerful mathematical tool used to find areas under curves, among other applications. It is particularly useful in this exercise to determine the area under the nitrogen concentration function, which helps us find the average concentration. Integration can be thought of as the process of summing an infinite number of infinitesimally small quantities, thus enabling us to calculate things like total distance, area, or in this case, the total nitrogen concentration along a transect.

Here's a quick introduction to integration:

• When you integrate a function, you are essentially finding its antiderivative.
• In practical terms, this means calculating the accumulated value or area under the curve represented by the function.
• In our exercise, this area corresponds to the total concentration of nitrogen along the transect from 0 to 10.

By understanding integration, you can not only solve this problem but apply similar methods to other real-world scenarios where accumulation needs to be calculated.

Linear Equation

A linear equation is a mathematical expression that represents a straight line when plotted on a graph. In this exercise, the nitrogen concentration along the transect is described by the linear equation \(y = 673.8 - 34.7x\).

Some key points to remember about linear equations include:

• They are of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• The slope \(m\) indicates how steep the line is and the direction of the line.
• In our equation, the slope \(-34.7\) suggests that nitrogen concentration decreases as you move further along the transect.
• The y-intercept \(673.8\) tells us the starting concentration when \(x = 0\).

Understanding these elements of a linear equation allows us to interpret the data more clearly, like predicting how concentration changes over distance.

Definite Integral

A definite integral is used to calculate the accumulation of quantities, such as areas, between two boundary points. In this context, it's applied to find the total concentration of nitrogen over a specific range of distances.

Here's how definite integrals work:

• The definite integral of a function \(\int_{a}^{b} f(x) \, dx\) calculates the net area under the curve \(f(x)\) from point \(a\) to point \(b\).
• It gives us a precise numerical value, unlike indefinite integrals which provide a general formula.
• By solving the definite integral for the given linear equation, we found the total concentration of nitrogen from the start to the end of the transect.

In summary, definite integrals are essential for quantifying total accumulation when dealing with continuous functions across a specific interval.

Transect Analysis

Transect analysis involves studying variations along a linear space to infer underlying trends or patterns. This technique is commonly used in environmental science to assess changes in soil composition, vegetation, or other ecological variables over a given distance.

In this exercise, transect analysis is applied as follows:

• The transect is a straight line, along which the concentration of nitrogen is measured at various points.
• By implementing a linear equation, we predict how soil nitrogen concentration changes with distance.
• The outcome helps in understanding environmental gradients and their potential impacts on local ecosystems.

Conducting a transect analysis allows researchers to collect systematic samples and derive accurate conclusions about spatial variation, providing vital insights into environmental and geographical studies.