Problem 8 - Applied Calculus

Question

The following table gives the US emissions, \(H(t),\) of hydrofluorocarbons, or "super greenhouse gasses," in teragrams equivalent of carbon dioxide, with \(t\) in years since \(2000 .^{7}\) (a) What are the units and meaning of \(\int_{0}^{10} H(t) d t ?\) (b) Estimate \(\int_{0}^{10} H(t) d t\) $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2002 & 2004 & 2006 & 2008 & 2010 \\ \hline H(t) & 7104 & 7022 & 7163 & 7159 & 7048 & 6822 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) Teragrams of CO2 over 10 years. (b) Approximately 27970 teragrams of CO2.

Step 1: Understanding the Units of Integration

The units of the integral \[\int_{0}^{10} H(t) \, dt\] represent the accumulation of hydrofluorocarbon emissions, measured in teragrams of carbon dioxide equivalent, over a period of 10 years from 2000 to 2010. This integral gives the total emissions over this time frame.

Step 2: Interpret the Meaning of the Integral

The integral \[\int_{0}^{10} H(t) \, dt\] provides a measure of the total emissions of hydrofluorocarbons in teragrams of CO2 equivalent accumulated over the first ten years of the 21st century. It can be interpreted as the area under the curve represented by \(H(t)\) from \(t = 0\) to \(t = 10\).

Step 3: Estimating the Integral Using Trapezoidal Rule

To estimate the integral, we can use the trapezoidal rule given the emission values at discrete points.The trapezoidal rule for estimating \[\int_{0}^{10} H(t) \, dt\] can be calculated as:\[\frac{b-a}{2n} \left[ f(a) + 2(f_1 + f_2 + \ldots + f_{n-1}) + f(b) \right]\]where \(a = 0\), \(b = 10\), \(n = 5\), and the value of \(f(t)\) at points are given by the table.Using given data:\[\int_{0}^{10} H(t) \, dt \approx \frac{10-0}{2 \times 5} \left[ 7104 + 2(7022 + 7163 + 7159 + 7048) + 6822 \right]\]Calculating inside the brackets sums to:\[7104 + 14044 + 6822= 27970\]Finally, multiplying by \(\frac{10}{10}\) gives us the estimate for \[\int_{0}^{10} H(t) \, dt \approx 27970\]Thus, the estimated total emission from 2000 to 2010 is approximately 27970 teragrams of CO2 equivalent.

Key Concepts

Hydrofluorocarbon Emissions

Hydrofluorocarbons (HFCs) are synthetic gases used in a variety of applications, such as refrigeration and air conditioning. They are considered "super greenhouse gases" because they have a high global warming potential (GWP), meaning they trap heat in the atmosphere much more effectively than carbon dioxide. Unlike naturally occurring gases, HFCs are manufactured and emitted directly by industrial activities.
These emissions are measured in teragrams of carbon dioxide equivalent, which indicates how much CO2 would cause the same amount of warming over a given time. This is a crucial measure, as it allows scientists and policymakers to evaluate the impact of various greenhouse gases on climate change. The measurement dates within a specific period, like from 2000 to 2010 in this exercise, help understand trends and impacts over time.

• Usage in multiple industries adds to their varied sources.
• Management and reduction of emissions are part of global climate agreements.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to estimate the definite integral of a function. In scenarios where we only know the function at specific points, like the hydrofluorocarbon emissions in the years 2000, 2002, 2004, etc., it proves very useful. This method approximates the area under the curve by dividing it into trapezoids instead of the exact, smooth curve of the function.
The formula for the Trapezoidal Rule is:

• \[\int_{a}^{b} f(t) \, dt \approx \frac{b-a}{2n} \left[ f(a) + 2(f_1 + f_2 + \ldots + f_{n-1}) + f(b) \right]\]

This method is applied step-by-step to each interval between the known points. The more trapezoids, the better the approximation, but even with fewer, it provides a decent estimate. Here, using the emission values every two years, we get an estimation of the total emissions over the full ten-year period.

Carbon Dioxide Equivalent

When discussing greenhouse gas emissions, the term **carbon dioxide equivalent (CO2e)** is frequently used. It helps compare the warming effects of different gases by converting them into the equivalent amount of CO2 that would have the same warming effect over a given time.
Each greenhouse gas has a different global warming potential, and thus, scientists use CO2e to simplify and standardize measurements. By doing so, they can provide a clear picture of a region's total contribution to global warming.

• HFCs, due to their chemically potent nature, often have much higher CO2e.
• This equivalence helps in both local and international climate policy formulation.
• The United States, as measured in this exercise, uses CO2e to convey the importance of reducing hydrofluorocarbon emissions.

This standard measure is crucial in climate science because it translates varied and complex data into a format that is more easily understood and actionable by everyone from scientists to policymakers.