Problem 31 - Applied Calculus

Question

The amount of pollution in a lake \(x\) years after the closing of a chemical plant is \(P(x)=100 / x\) tons (for \(x \geq 1\) ). Find the average amount of pollution between 1 and 10 years after the closing.

Step-by-Step Solution

Answer

The average pollution between years 1 and 10 is approximately \(\frac{100\ln(10)}{9}\) tons.

Step 1: Understanding the Problem

We need to find the average amount of pollution in the lake between 1 year and 10 years after the closing of a chemical plant. The pollution is given as a function of time: \[ P(x) = \frac{100}{x} \] where \(x\) is the number of years after the plant's closure.

Step 2: Integrating the Function

To find the average pollution over a time interval \([a, b]\), we use the integral of the pollution function over that interval:\[ \int_{1}^{10} \frac{100}{x} \, dx \]This expression will give us the total pollution from year 1 to year 10.

Step 3: Finding the Antiderivative

The antiderivative of \( \frac{100}{x} \) is:\[ 100 \int \frac{1}{x} \, dx = 100 \ln |x| + C \] where \(C\) is the constant of integration.

Step 4: Evaluating the Definite Integral

Calculate the definite integral from 1 to 10:\[ \int_{1}^{10} \frac{100}{x} \, dx = 100[\ln(10) - \ln(1)] = 100\ln(10) \] Since \( \ln(1) = 0 \), the evaluation simplifies to \( 100\ln(10) \).

Step 5: Calculating the Average Pollution

The average pollution over the interval \([1, 10]\) is found by dividing the total pollution by the length of the interval. This is given by the formula:\[ \text{Average} = \frac{1}{10 - 1} \int_{1}^{10} \frac{100}{x} \, dx = \frac{1}{9} \cdot 100\ln(10) \]Simplify the expression further:\[ \text{Average} = \frac{100\ln(10)}{9} \]

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of integration. It deals with calculating the area under curves and extends to determining various physical quantities like displacement, area, volume, and more.

To understand integral calculus, it's important to first grasp the fundamentals of a function. A function describes a relationship between two variables—often representing data or trends within a specific context that can be visualized as a graph. For instance, in our exercise, the function represents pollution over time.

• The integral of a function calculates the accumulation of values that the function takes over a certain interval. This is visually represented as the area under the curve.
• There are two types of integrals: indefinite and definite. An indefinite integral provides a general formula for antiderivatives, while a definite integral has limits attached, providing a specific numeric value.

In application, integral calculus allows us to solve practical problems such as finding total pollution over a period. This is relevant when determining the effectiveness of environmental policies or understanding pollution impacts over time.

Pollution Modeling

Pollution modeling is a method used to estimate and understand pollutant levels in the environment—such as air, water, or soil—over time. It's essential for assessing environmental impacts and making informed policy decisions.

In our scenario, pollution modeling involves using a mathematical function to represent how pollution from a chemical plant declines over time since the plant's closure. This model is expressed as:

• Function: \( P(x) = \frac{100}{x} \)
• Here, \(x\) represents the number of years after the plant shutdown.
• The function descends as \(x\) increases, illustrating the decrease in pollution as time goes on.

Understanding this function is critical for analyzing trends in pollution data and anticipating future pollution levels. By predicting how a pollutant’s concentration decreases, policymakers can effectively gauge whether current environmental interventions are successful or if additional measures are required.

Definite Integrals

Definite integrals are crucial in calculus for determining the precise accumulation of a quantity—such as pollution—over a specified interval. Unlike indefinite integrals which give a broad family of solutions, definite integrals calculate exact values.

• Definite integrals are written with upper and lower limits indicating the interval of interest, shown as \( \int_{a}^{b} f(x) \, dx \).
• This process computes the net area under the curve between points \(a\) and \(b\) on the \(x\)-axis.
• The result includes any areas below the \(x\)-axis as negative, reflecting a decrease in value (if applicable).

For our exercise, the definite integral \( \int_{1}^{10} \frac{100}{x} \, dx \) provides the total pollution from year one to year ten.

Calculating this definite integral allows us to understand the total environmental impact during this period. To find the average pollution, you then take this result and divide it by the number of years, which matches the formula for finding mean values across a timeline.