Problem 1 - Applied Calculus

Question

The following table gives the emissions, \(E\), of nitrogen oxides in millions of metric tons per year in the US. Let \(t\) be the number of years since 1970 and \(E=f(t)\). (a) What are the units and meaning of \(\int_{0}^{30} f(t) d t\) ? (b) Estimate \(\int_{0}^{30} f(t) d t\). $$ \begin{array}{l|l|l|l|l|l|l|l} \hline \text { Year } & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 \\ \hline E & 26.9 & 26.4 & 27.1 & 25.8 & 25.5 & 25.0 & 22.6 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

(a) The integral estimates total emissions (in million metric tons). (b) The estimated emissions are about 587 million metric tons.

Step 1: Understand the Integral's Meaning

The integral \( \int_{0}^{30} f(t) \, dt \) represents the total accumulation of nitrogen oxide emissions over 30 years from 1970 to 2000. In this context, the units would be in millions of metric tons, as each year's emissions are measured in millions of metric tons.

Step 2: Use Trapezoidal Rule for Estimation

To estimate \( \int_{0}^{30} f(t) \, dt \), we will apply the trapezoidal rule. The formula is given by \( \int_{a}^{b} f(t) \, dt \approx \frac{b-a}{2n} [f(t_0) + 2f(t_1) + 2f(t_2) + \ldots+ 2f(t_{n-1}) + f(t_n)] \), where \( n \) is the number of intervals. Here, \( n=6 \) with the differences \( t=5 \) years between each two points.

Step 3: Apply the Values into the Formula

Compute using the values from the table: Year values correspond to \( f(t) \): \[ \frac{5}{2} \left( 26.9 + 2(26.4 + 27.1 + 25.8 + 25.5 + 25.0) + 22.6 \right) = \frac{5}{2} \times (26.9 + 52.8 + 54.2 + 51.6 + 50.5 + 44.2 + 22.6) = \frac{5}{2} \times 234.8 = 587 \text{ million metric tons} \]

Step 4: Interpret the Result

The estimated total emissions of nitrogen oxides over the 30-year period is approximately 587 million metric tons, derived using the trapezoidal approximation method.

Key Concepts

Emissions Estimation

Estimating emissions is a crucial part of understanding environmental impacts over time. This involves calculating the total amount of pollutants, such as nitrogen oxides, released into the atmosphere across a specific period. In this case, the problem focuses on nitrogen oxide emissions in the US from 1970 to 2000. To find this total, we use integral calculus. Here, the integral \( \int_{0}^{30} f(t) \, dt \) represents the sum of emissions over 30 years, where \( E = f(t) \) indicates emissions for each year. Using the units effectively:

• Years since 1970 are marked simply as \( t \) (due to the sequential nature of years).
• Emissions, \( E \) , are measured in millions of metric tons.

Understanding this helps grasp the size and growth of emissions over decades, aiding in decision-making for environmental policies.

Trapezoidal Rule

The trapezoidal rule is a numerical method used to approximate the definite integral of a function. It simplifies complex calculations by breaking down the area under a curve into smaller, manageable trapezoids. This method is particularly useful when working with data points from real-life scenarios, as it provides a straightforward approximation technique. For this specific problem, where emissions are examined every five years, the trapezoidal rule is ideal because:

• It utilizes the given values directly: \( E = f(t) \) from years \( 1970 \) through \( 2000 \).
• The interval \( n \) is \( 6 \) (since there are six intervals).
• The formula used provides an approximate total: \[ \int_{a}^{b} f(t) \, dt \approx \frac{b-a}{2n} [f(t_0) + 2f(t_1) + \ldots + f(t_n)] \]This formula calculates the sum of each trapezoid's area and multiplies it by the height (the interval length divided by \( 2 \)).

This method eases the estimation of nitrogen oxide emissions over the specified period, reaching an approximate total of \( 587 \) million metric tons.

Nitrogen Oxides

Nitrogen oxides (\( NO_x \) ) are a group of gases that significantly contribute to air pollution. They mostly originate from vehicle emissions, industrial processes, and electrical power plants. Understanding their emissions is vital since they have extensive environmental and health impacts. Here's why nitrogen oxides are important:

• Contribute to the formation of smog and acid rain.
• Cause respiratory problems and other health issues in humans.
• Influence climate change by affecting the atmospheric balance of gases.

In the problem reviewed, interpreting the total emissions over 30 years helps understand how policies and technologies might influence pollution levels. This data also aids in developing future strategies to reduce emissions and mitigate negative effects on the environment and human health. Analyzing such data allows stakeholders to better document changes over time and influences policy-making for cleaner air initiatives.