Problem 2 - Calculus Single Variable

Question

Use the first part of the Fundamental Theorem of Calculus together with the ideas of Examples 1 and 2 to evaluate the definite integrals in Exercises \(1-10\) $$ \int_{-1}^{0} e^{-x} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \( e - 1 \).

Step 1: Recognize the Integral Form

The given integral is \( \int_{-1}^{0} e^{-x} \, dx \). The integral is in the form \( \int_{a}^{b} f(x) \, dx \), where \( f(x) = e^{-x} \), \( a = -1 \), and \( b = 0 \).

Step 2: Identify the Antiderivative

To evaluate the integral using the Fundamental Theorem of Calculus, we need the antiderivative of \( f(x) = e^{-x} \). The antiderivative of \( e^{-x} \) is \( -e^{-x} \).

Step 3: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( F(x) \) is an antiderivative of \( f(x) \), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \). Here, \( F(x) = -e^{-x} \), so we calculate \( F(0) - F(-1) \).

Step 4: Evaluate at the Upper Limit

Calculate \( F(0) = -e^{-0} = -1 \).

Step 5: Evaluate at the Lower Limit

Calculate \( F(-1) = -e^{-(-1)} = -e^{1} = -e \).

Step 6: Calculate the Definite Integral

Substitute the values into the theorem: \( F(0) - F(-1) = -1 - (-e) = -1 + e \).

Step 7: Simplify the Result

The final result is \( e - 1 \).

Key Concepts

Definite Integrals

Definite integrals help us find the accumulated area under a curve between two specific points on a graph. This is a core aspect of calculus, allowing for the calculation of total accumulated quantities. A definite integral is represented as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the lower and upper limits respectively. This setup tells us the interval over which we are finding the area.
This area takes into account the shape and slope of the curve created by the function \( f(x) \) between points \( a \) and \( b \).

• The value may be positive, negative, or zero.
• Positive if the area is above the x-axis, and negative if below.
• Zero can indicate a balance of areas above and below the x-axis.

The problem given is a definite integral \( \int_{-1}^{0} e^{-x} \, dx \), asking for the total area between \( x = -1 \) and \( x = 0 \), moving along the curve of \( e^{-x} \). These calculations yield an exact area measure for those x-values, illustrating how definite integrals encapsulate the full picture of changes within a specified range.

Antiderivatives

In calculus, an antiderivative is a function whose derivative gives the original function back. Essentially, it's the reverse process of differentiation. When working with definite integrals, finding the antiderivative is crucial. For example, if \( f(x) = e^{-x} \), then its antiderivative is \( F(x) = -e^{-x} \). The fundamental theorem of calculus tells us how to use antiderivatives to evaluate integrals.

• Given \( F(x) \) as an antiderivative of \( f(x) \), the definite integral \( \int_{a}^{b} f(x) \, dx \) equals \( F(b) - F(a) \).
• This provides a simple method to compute areas or accumulated changes quickly.

Understanding and identifying antiderivatives is a key task when solving integrals, making them indispensable for calculating definite integrals.

Exponential Functions

Exponential functions are functions of the form \( f(x) = a^{x} \), where the base \( a \) is a constant and the exponent \( x \) is a variable. They grow or decay at rates proportional to their current value, making them applicable in modeling real-world phenomena like population growth and radioactive decay.
In mathematics, the function \( e^{x} \) is a specific exponential function where the base \( e \) is Euler's number, approximately 2.718. This function exhibits unique properties:

• The derivative of \( e^{x} \) is \( e^{x} \); it remains unchanged upon differentiation.
• Conversely, the antiderivative of \( e^{x} \) is also \( e^{x} \).

However, when the function is \( e^{-x} \), its shape is reflected across the y-axis, demonstrating exponential decay. Calculating the integral of \( e^{-x} \) involves finding the antiderivative \( -e^{-x} \) as in our given problem, crucial in evaluating the definite integral using the fundamental theorem of calculus.