Problem 61 - Applied Calculus

Question

Evaluate each definite integral. $$ \int_{-1}^{1} 5 e^{-x} d x $$

Step-by-Step Solution

Answer

The definite integral evaluates to approximately 11.752.

Step 1: Recognize the Form of the Integral

Identify the integral as a definite integral of the form \( \int_{a}^{b} f(x) \, dx \), where \( f(x) = 5e^{-x} \), \( a = -1 \), and \( b = 1 \).

Step 2: Integrate the Function

Integrate the function \( f(x) = 5e^{-x} \). The antiderivative of \( e^{-x} \) is \( -e^{-x} \). Thus, the antiderivative of \( 5e^{-x} \) is \( -5e^{-x} \).

Step 3: Apply the Fundamental Theorem of Calculus

Use the Fundamental Theorem of Calculus to evaluate the definite integral. Substitute the upper limit \( x = 1 \) and lower limit \( x = -1 \) into the antiderivative, then find the difference: \[ \int_{-1}^{1} 5e^{-x} \, dx = \left[ -5e^{-x} \right]_{-1}^{1} = -5e^{-1} - (-5e^{1}). \]

Step 4: Simplify the Expression

Simplify the expression: \[ -5e^{-1} + 5e^{1} = 5(e^{1} - e^{-1}) = 5 \left( e - \frac{1}{e} \right). \]

Step 5: Calculate the Result

Calculate the result as precisely as needed. Since \( e \approx 2.71828 \), compute: \[ 5 \left( 2.71828 - \frac{1}{2.71828} \right) \approx 5 (2.71828 - 0.367879) = 5(2.350401) \approx 11.752. \]

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a cornerstone of calculus, linking the concepts of differentiation and integration. It tells us how to solve a definite integral using antiderivatives. In simple terms, it provides a way to calculate the area under a curve when you know an antiderivative of the function involved.

When we want to evaluate a definite integral, such as \( \int_{a}^{b} f(x) \, dx \), the theorem states that if \( F \) is an antiderivative of \( f \), then:

• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

This means we find the antiderivative, evaluate it at the upper limit, evaluate it at the lower limit, and subtract the two results.

In the exercise, after finding the antiderivative as \( -5e^{-x} \), we applied this theorem by substituting the upper limit \( x = 1 \) and the lower limit \( x = -1 \), then computing \( -5e^{-1} - (-5e^{1}) \). This is how we utilize the fundamental theorem to find exact values of integrals.

Exponential Function Integration

Integrating exponential functions, like \( e^{-x} \), often appears in calculus problems. It is crucial to understand the impact of the base \( e \), the natural exponential function. The natural exponential function has the special property where its derivative is itself.

For integration, the function \( e^{-x} \) results in an antiderivative of \( -e^{-x} \), because integrating reverses differentiation. For a scaled function such as \( 5e^{-x} \), integration involves maintaining the constant:

• The integral of \( 5e^{-x} \) leads to an antiderivative of \( -5e^{-x} \).

This process involves recognizing the structure of the exponential function and understanding how constants are preserved during integration.

Recognizing these patterns helps simplify integration tasks, especially when encountering different exponential expressions. The simplicity of exponential functions makes them essential components of calculus, frequently surfacing in scientific and engineering applications.

Antiderivative Calculation

Calculating antiderivatives is an essential skill in solving definite integrals. It involves finding a function whose derivative is the given function. An antiderivative acts as 'the opposite' of a derivative.

When finding the antiderivative of expressions such as \( 5e^{-x} \), we need to know how individual terms contribute. The constant 5, associated with \( e^{-x} \), is handled by multiplying the antiderivative of the exponential part, which is \( -e^{-x} \). Thus, bringing us to the antiderivative \( -5e^{-x} \).

Key steps to remember while calculating antiderivatives:

• Recognize the basic form of the function (e.g., exponential, polynomial)
• Use known formulas to find an antiderivative for the base function
• Apply any constants present directly to the antiderivative

Learning to quickly identify and manipulate these elements is critical for smooth integration and subsequent application of the Fundamental Theorem of Calculus. Familiarity with common derivatives translates directly into ease of finding antiderivatives.