Problem 45 - Calculus

Question

Evaluate the integrals. $$ \int_{0}^{1} e^{2 x} d x $$

Step-by-Step Solution

Answer

\(\frac{1}{2} (e^{2} - 1)\)

Step 1: Identify the Integration Formula

We need to evaluate the integral \( \int_{0}^{1} e^{2x} \, dx \). This is a definite integral involving the exponential function \( e^{2x} \).

Step 2: Perform the Integration

To integrate \( e^{2x} \), we use the basic integral formula for exponentials: \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). Here, \( a = 2 \), so we have\[\int e^{2x} \, dx = \frac{1}{2} e^{2x} + C.\]

Step 3: Apply the Limits of Integration

Since it is a definite integral, we apply the limits from 0 to 1:\[\left[ \frac{1}{2} e^{2x} \right]_{0}^{1} = \frac{1}{2} e^{2 \cdot 1} - \frac{1}{2} e^{2 \cdot 0}.\]

Step 4: Calculate the Value

Now, evaluate the expression:\[\frac{1}{2} e^{2} - \frac{1}{2} e^{0} = \frac{1}{2} e^{2} - \frac{1}{2} \times 1 = \frac{1}{2} (e^{2} - 1).\]Thus, the value of the integral is \( \frac{1}{2} (e^{2} - 1) \).

Key Concepts

Exponential Functions

Exponential functions are a key concept in calculus and mathematics as a whole. An exponential function has the form \( e^{ax} \), where \( e \) is a mathematical constant approximately equal to 2.71828, and \( a \) is a constant which determines the growth rate of the function. These functions are unique because their rate of growth is proportional to their current value. This means that as you increase \( x \), the value of the function grows rapidly.
Exponential functions are widely used to model real-world phenomena, including population growth, radioactive decay, and continuously compounded interest. In the context of integration, understanding the behavior of exponential functions helps you assess how the area under their curve changes with different limits of integration.
Exponential functions are commonly denoted by \( e \) for the base, often seen in forms like \( e^{2x} \). Integrals involving such functions are solved using specific techniques adapted for these rapidly changing curves.

Basic Integration Formulas

When it comes to integrating functions like \( e^{2x} \), knowledge of basic integration formulas is essential. The specific formula for integrating an exponential function is \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \), where \( C \) represents the constant of integration for indefinite integrals.
This formula shows us how to handle the multiplication factor \( a \) in the exponent. For example, if we want to integrate \( e^{2x} \), we use the formula setting \( a = 2 \). This gives us \( \frac{1}{2} e^{2x} + C \).
Understanding basic integration formulas allows you to efficiently tackle a wide range of integration problems, and adjusting the formula for different values of \( a \) ensures you can adapt to more complex exponential functions. This fundamental knowledge sets the stage for evaluating more complex definite integrals involving exponential growth.

Limits of Integration

The limits of integration define the interval over which integration is performed. In a definite integral like \( \int_{0}^{1} e^{2x} \, dx \), these limits are 0 and 1, which specify the starting and ending points on the x-axis.
When calculating definite integrals, after finding the antiderivative, you evaluate the resulting expression at the upper limit and subtract the evaluation at the lower limit. Applying this to exponential functions involves substituting the limits into the antiderivative expression. For example, with \( \frac{1}{2} e^{2x} \), we evaluate as \( \left[ \frac{1}{2} e^{2x} \right]_{0}^{1} = \frac{1}{2} e^{2 \times 1} - \frac{1}{2} e^{2 \times 0} \).
The application of limits of integration ensures precision and allows the computation of the exact area under the curve, giving the final solution \( \frac{1}{2} (e^{2} - 1) \). This process turns abstract functions into concrete numerical results, revealing insights about the function's behavior over specified intervals.