Problem 16 - Applied Calculus

Question

Using the Fundamental Theorem, evaluate the definite integrals in problem exactly. $$ \int_{0}^{1} 2 e^{x} d x $$

Step-by-Step Solution

Answer

The exact value is \( 2e - 2 \).

Step 1: Identify the Function and Limits

The integral we need to evaluate is \( \int_{0}^{1} 2e^{x} \, dx \). Here, the function is \( f(x) = 2e^{x} \), and the limits of integration are from 0 to 1.

Step 2: Find the Antiderivative

To evaluate the definite integral, we first find the antiderivative of \( f(x) = 2e^{x} \). The antiderivative of \( e^{x} \) is \( e^{x} \), so the antiderivative of \( 2e^{x} \) is \( 2e^{x} \) because the constant factor remains unchanged.

Step 3: Apply the Fundamental Theorem

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, \( F(x) = 2e^{x} \), \( a = 0 \), and \( b = 1 \).

Step 4: Evaluate the Antiderivative at the Upper Limit

Compute \( F(1) = 2e^{1} = 2e \).

Step 5: Evaluate the Antiderivative at the Lower Limit

Compute \( F(0) = 2e^{0} = 2 \), since \( e^{0} = 1 \).

Step 6: Subtract the Values

Subtract the value of the antiderivative at the lower limit from the value at the upper limit: \( F(1) - F(0) = 2e - 2 \).

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus representing the accumulation of quantities. It is used to find the area under a curve between two points. In mathematical terms, it calculates the integral of a function over a specified interval.
For the exercise, the definite integral is represented as:\[\int_{0}^{1} 2e^{x} \, dx\]Here, 0 and 1 are the limits of integration, indicating where the calculation starts and ends along the x-axis.
The definite integral transforms the area under the curve of the function between these limits into a single numerical value. This value is often used to calculate physical quantities like distance or total growth. The outcome is determined by evaluating the antiderivative at the upper limit and subtracting the value of the antiderivative at the lower limit.

Antiderivative

The concept of an antiderivative refers to reversing the process of differentiation. It allows us to find a function whose derivative is the given function. This is crucial in solving definite integrals.
In the exercise, we find the antiderivative of \( f(x) = 2e^{x} \). To do this, we need to consider both the constant and the exponential component:

• The derivative of \( e^{x} \) is itself, \( e^{x} \).
• Therefore, the antiderivative of \( 2e^{x} \) is simply \( 2e^{x} \).

Antiderivatives are key to using the Fundamental Theorem of Calculus, which connects differentiation and integration. By knowing the antiderivative, we can evaluate the definite integral by simply substituting the limits of integration.

Exponential Function

The exponential function is a mathematical function denoted as \( e^{x} \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
This function has special properties:

• Its rate of growth is proportional to its current value, making it rise steeply as \( x \) increases.
• The exponential function is its own derivative and antiderivative.

This characteristic simplifies solving integrals and differential equations involving exponential functions. In this problem, we deal with \( 2e^{x} \), where the "2" multiplies the function \( e^{x} \) but doesn't affect its fundamental properties. Understanding these properties helps in predicting how the function behaves and determines the approach when finding antiderivatives.

Step-by-Step Solution

Following a step-by-step solution is essential for breaking down complex problems into manageable parts. This approach helps in applying mathematical rules systematically.
In the featured problem, the process begins by identifying the integral and its limits. Calculating the antiderivative is the next critical step. The Fundamental Theorem of Calculus then guides us to:

• Evaluate the antiderivative at the upper limit.
• Compute it at the lower limit.
• Subtract these results.

For the integral \( \int_{0}^{1} 2e^{x} \, dx \), this method results in \( 2e - 2 \), providing a concrete understanding of how to handle definite integrals analytically. This systematic approach is invaluable when tackling more complex integrals or different types of functions.