Problem 7 - Applied Calculus

Question

Annual coal production in the US (in billion tons per year) is given in the table. \(^{6}\) Estimate the total amount of coal produced in the US between 1997 and \(2009 .\) If \(r=f(t)\) is the rate of coal production \(t\) years since 1997, write an integral to represent the \(1997-2009\) coal production. $$\begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 1997 & 1999 & 2001 & 2003 & 2005 & 2007 & 2009 \\ \hline \text { Rate } & 1.090 & 1.094 & 1.121 & 1.072 & 1.132 & 1.147 & 1.073 \\\ \hline \end{array}$$

Step-by-Step Solution

Answer

Approx. 6.65 billion tons of coal were produced in the US between 1997 and 2009.

Step 1: Understand the Given Data

The table shows the rate of coal production in billion tons per year for selected years between 1997 and 2009. The rate in each year is represented by the function \( r = f(t) \), where \( t \) is the number of years since 1997.

Step 2: Set Up the Integral Expression

We want to find the total amount of coal produced from 1997 to 2009. This is given by the integral of the rate of coal production over this period:\[\int_{0}^{12} f(t) \, dt\] The limits of integration are from \( t = 0 \) (year 1997) to \( t = 12 \) (year 2009).

Step 3: Estimate Total Coal Production Using Trapezoidal Rule

Since we only have data for specific years, we can estimate the integral using the trapezoidal rule. The trapezoidal rule approximation is:\[\int_{0}^{12} f(t) \, dt \approx \frac{1}{2}(r_0 + 2r_2 + 2r_4 + 2r_6 + 2r_8 + 2r_{10} + r_{12})\]where \( r_0, r_2, ..., r_{12} \) are the rates of coal production for the years 1997, 1999, ..., 2009 respectively.

Step 4: Calculate the Trapezoidal Sum

Substituting the given rates:\[\frac{1}{2}(1.090 + 2(1.094) + 2(1.121) + 2(1.072) + 2(1.132) + 2(1.147) + 1.073)\]\[= \frac{1}{2}(1.090 + 2.188 + 2.242 + 2.144 + 2.264 + 2.294 + 1.073)\]\[= \frac{1}{2}(13.295)\]\[= 6.6475\]Thus, the estimated total coal production between 1997 and 2009 is approximately 6.6475 billion tons.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of definite integrals, especially when we have discrete data points as opposed to a continuous function that can be integrated analytically. In essence, it works by dividing the area under a curve into a series of trapezoids rather than perfect rectangles, which provides a more accurate approximation in many cases.
Here's how it works:

• Imagine tracing the top of a trapezoid along the curve we want to integrate.
• Each 'trapezoid' has a base along the x-axis, and two sides running up to the corresponding points on the curve.
• The formula for n trapezoids is:
  \[\int_{a}^{b} f(t) \, dt \approx \frac{1}{2}(f(a) + 2f(t_1) + 2f(t_2) + \ldots + 2f(t_{n-1}) + f(b))\delta t\]
  where \( \delta t \) is the length of each segment along the x-axis.
• This method is particularly useful when the function in question is not easily integrable using standard calculus techniques, allowing for a practical and efficient solution.

The use of the Trapezoidal Rule in the coal production problem helps estimate the integral of the function representing the rate of coal production over several discrete years.

Coal Production Estimation

To estimate the coal production from 1997 to 2009, we need to calculate the total amount of coal produced by using the given rate of production data. This requires integrating the rate function over the given time period.
Here's how the estimation process works:

• The function \( f(t) \) signifies the rate of coal production (in billion tons per year).
• We have recorded values for specific years: 1997, 1999, 2001, 2003, 2005, 2007, and 2009.
• By applying the Trapezoidal Rule, we use these rate values to shape our approximation of the definite integral that represents the total production.
• Substituting and calculating gives the estimated value: the sum of products of these trapezoidal areas gives us approximately 6.6475 billion tons.

The calculated approximation provides an insight into how much coal has been produced within this 12-year span, which would be crucial for historical analysis and future predictions in coal industry planning.

Definite Integral

A definite integral can be thought of as a tool that helps us find the total accumulation of a quantity, which in our exercise is the total coal production.
Here's a deeper understanding of the concept:

• A definite integral \( \int_{a}^{b} f(t) \, dt \) calculates the accumulation of \( f(t) \) between \( t = a \) and \( t = b \).
• This is not just about finding an area under a curve, but understanding the total accumulation of change over an interval, such as total production over years.
• In this context, the rate function \( f(t) \) is integrated over \( t = 0 \) to \( t = 12 \), showing the sum of coal produced from the start year to the final year—here 1997 to 2009.
• When applying this concept, using numerical methods like the Trapezoidal Rule helps when the integral cannot be evaluated analytically, especially with empirical data.

Thus, the definite integral and its numerical approximation lays the foundation for practical estimation problems, like determining total production, which has immediate real-world implications.