Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the definite integral. $$\int_{0}^{1} e^{-x} d x$$

Step-by-Step Solution

Answer

The definite integral \(\int_{0}^{1} e^{-x} dx\) can be evaluated by finding the antiderivative of \(e^{-x}\), which is \(-e^{-x}\). Applying the Fundamental Theorem of Calculus, we get \(-e^{-1} - (-e^{-0})\), which simplifies to \(1 - \frac{1}{e} \approx 0.632\).

Step 1: Finding the antiderivative

To find the antiderivative of \(e^{-x}\), we can use the following formula for the exponential function: \[ \int e^{ax} dx = \frac{1}{a} e^{ax} + C \] where, a and C are constants. In our case, \(a = -1\). So, \[ \int e^{-x} dx = \frac{1}{-1} e^{-x} + C = -e^{-x} + C \] Step 2: Apply the Fundamental Theorem of Calculus

Step 2: Applying the Fundamental Theorem of Calculus

Now we will evaluate the definite integral by applying the Fundamental Theorem of Calculus, which states: \[ \int_{a}^{b} f(x) dx = F(b) - F(a) \] where F(x) is the antiderivative of f(x). In our case, \(a = 0\), \(b = 1\), and \(F(x) = -e^{-x}\). So, \[ \int_{0}^{1} e^{-x} dx = -e^{-1} - (-e^{-0}) \] Step 3: Evaluate and simplify the expression

Step 3: Evaluate and simplify

Now, we will compute the definite integral using the expression we derived: \[ -e^{-1} - (-e^{-0}) = - \frac{1}{e} -(-1) \] We can simplify this expression: \[ - \frac{1}{e} + 1 \approx 1 - 0.368 \approx 0.632 \] Thus, the value of the definite integral is approximately 0.632.

Key Concepts

Antiderivative

Antiderivatives are the inverse operation of taking derivatives. In essence, it's the process of finding the original function for which the derivative is given. For students working with exponential functions in calculus, understanding antiderivatives is a stepping stone to solving integrals.

Working with Exponentials

When dealing with the exponential function, particularly with a negative exponent as in the exercise \( e^{-x} \), the antiderivative is found by applying the general formula \[ \int e^{ax} dx = \frac{1}{a} e^{ax} + C \] where \( a \) is the constant coefficient of \( x \) and \( C \) represents the constant of integration. It's crucial here to note that when dealing with definite integrals, the constant \( C \) cancels out when we evaluate the integral within the limits, making the process slightly easier.

For the given integral, \( \int e^{-x} dx \), identifying \( a=-1 \) allows us to find the antiderivative \( -e^{-x} \) fairly straightforwardly. This antiderivative is the building block for the next step: evaluating the definite integral using the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration, two fundamental concepts in calculus. This theorem is divided into two parts, but for the purpose of solving definite integrals, the second part is particularly relevant.

Application in Definite Integrals

The second part of the Theorem states that if \( F \) is the antiderivative of a continuous function \( f \) on an interval \[a,b\], then \[ \int_{a}^{b} f(x) dx = F(b) - F(a) \]. This essentially means that to find the value of a definite integral, one only needs to evaluate the antiderivative at the upper and lower limits of integration and then subtract the latter from the former.

In our exercise, after finding the antiderivative \( -e^{-x} \), we apply the theorem by evaluating this function at \( x=1 \) and \( x=0 \) to find the integral's value over the interval \[0,1\]. This yields a result that effectively captures the 'area under the curve' of the original function on that interval.

Exponential Function Integration

The integration of exponential functions often involves base \( e \) due to its unique properties in calculus, particularly the natural logarithm's base. The exponential function is defined as \( e^{x} \), and when \( x \) has a negative coefficient as in \( e^{-x} \), the integration process involves a flip in sign for the final antiderivative.

Simplifying Results

Once the antiderivative is determined, as with our function \( -e^{-x} \), the next step in evaluating the definite integral of this function is to plug in the limits of integration, simplify the resulting expression, and then calculate the numerical value if needed. In many real-world applications, it is the relationship that these functions describe that is of far more interest than the actual numeral.

For example, while calculating \( \int_{0}^{1} e^{-x} dx \), after simplification, we observe that the integral evaluates to \( - \frac{1}{e} + 1 \), which approximately equals 0.632. This numerical value has a concrete interpretation in the context it is used in, but understanding the step-by-step integration process is critical for students.