Problem 36 - Applied Calculus

Question

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=6 e^{2 x} \text { from } x=0 \text { to } x=2 $$

Step-by-Step Solution

Answer

The area under \(f(x) = 6e^{2x}\) from \(x=0\) to \(x=2\) is \(3(e^{4} - 1)\).

Step 1: Identify the Function and Limits

The given function is \( f(x) = 6 e^{2x} \) and the task is to find the area under this curve between \( x = 0 \) and \( x = 2 \). These values will serve as the limits of our definite integral.

Step 2: Set Up the Definite Integral

To find the area under the curve, we need to set up the definite integral. The integral is expressed as \[ \int_{0}^{2} 6e^{2x} \, dx \]. This integral will compute the total area under the function \( f(x) \) from \( x = 0 \) to \( x = 2 \).

Step 3: Integrate the Function

We need to find the antiderivative of \( 6e^{2x} \). The antiderivative is found using the formula for integration of exponential functions: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). Thus, the antiderivative of \( 6e^{2x} \) is \( \frac{6}{2} e^{2x} = 3e^{2x} \).

Step 4: Apply the Limits of Integration

Substitute the limits into the antiderivative. This results in \[ \left. 3e^{2x} \right|_{0}^{2} = 3e^{4} - 3e^{0} \].

Step 5: Calculate the Value

Calculate \(3e^{4} - 3e^{0}\). Since \(e^{0} = 1\), the expression simplifies to \(3e^{4} - 3\).

Step 6: Final Calculation

Calculate the numerical value or leave it in terms of \(e\): \(3(e^{4} - 1)\). This is the area under the curve from \(x=0\) to \(x=2\). For a numerical approximation, use a calculator to find \(e^{4}\) and compute the result.

Key Concepts

Area Under a Curve

When we talk about finding the area under a curve, we refer to the space that lies between a curve described by a function and the horizontal axis, across a specific interval. This is a common concept in calculus as it allows us to determine accumulated quantities, like distance, volume, or total revenue, over time.

To find this area precisely, we use the definite integral. The "definite" part means we are calculating the integral from a specific starting point to an ending point, which are often represented as limits. For instance, in the example given with the function \( f(x) = 6e^{2x} \), our limits are from \( x=0 \) to \( x=2 \).

• Set up the definite integral as \( \int_{0}^{2} 6e^{2x} \, dx \)
• Evaluate this integral to compute the exact area under the curve
  

By working out this integral, we obtain the total accumulation under the curve between these two x-values. This approach provides exact results, which can then be rounded or approximated using tools like calculators.

Exponential Function

Exponential functions are special mathematical functions where a constant base is raised to a variable exponent. They are widely represented in the form \( f(x) = a \cdot e^{bx} \), where \( e \) is approximately 2.71828, known as Euler's number, and is the base of the natural logarithm.

In our current problem, the function \( f(x) = 6e^{2x} \) is an exponential function. Here, the base is \( e \), and the exponent is a function of \( x \), specifically \( 2x \). This means the function grows, or increases, very rapidly as \( x \) increases. The coefficient 6 stretches the function vertically making it larger compared to \( e^{2x} \) alone.

• Exponential growth occurs because the function's rate of change increases as the value of \( x \) increases
• This type of function is crucial in modeling real-world processes like population growth, radioactive decay, and calculating compound interest

Exponential functions have unique properties and understanding them is fundamental to grasping how they are integrated and how they behave over specific intervals.

Antiderivative

In calculus, finding an antiderivative means determining the original function that resulted in a given derivative. This is the reverse process of differentiation, so it’s also called integration. The antiderivative is crucial when solving definite integrals, as it helps in calculating areas under curves.

For the function \( f(x) = 6e^{2x} \), we find its antiderivative by reversing the differentiation rules applied to exponential functions. The general rule for integrating \( e^{ax} \) is:

• \( \int e^{ax} \, dx = \frac{1}{a}e^{ax} + C \)

Applying this rule gives us the antiderivative of \( 6e^{2x} \) as \( 3e^{2x} \), noting how the multiplication by \( \frac{1}{2} \) from the coefficient "2" in the exponent balances the differentiation operation.

This knowledge of antiderivatives is vital because it is the step that leads us to solve the integration process, allowing us to compute areas under curves or solve a variety of practical problems involving accumulation. The constant \( C \) represents the "constant of integration" and becomes irrelevant in definite integrals since it cancels out when evaluating the boundaries.