Problem 10 - Applied Calculus

Question

Using the Fundamental Theorem, evaluate the definite integrals in Problems \(1-20\) exactly. $$\int_{0}^{1} 2 e^{x} d x$$

Step-by-Step Solution

Answer

The integral evaluates to \(2e - 2\).

Step 1: Identify the Integral and Setup

We begin with the definite integral \( \int_{0}^{1} 2e^{x} \, dx \). This requires finding the antiderivative of the integrand \( 2e^{x} \).

Step 2: Find the Antiderivative

The antiderivative of \( 2e^{x} \) is \( 2e^{x} \), since the derivative of \( e^{x} \) is itself and the constant multiple remains unchanged.

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, evaluate the antiderivative at the upper limit and the lower limit of the integral and calculate the difference:\[\left[ 2e^{x} \right]_{0}^{1} = 2e^{1} - 2e^{0}.\]

Step 4: Simplify the Expression

Substitute and simplify using \( e^{1} = e \) and \( e^{0} = 1 \):\( 2e - 2 \).

Step 5: Final Calculation

Compute the simplified expression: 2e - 2.

Key Concepts

Definite Integrals

Definite integrals are a crucial concept in calculus. They help you find the exact total of a function within a specific range. This is different from indefinite integrals, which don't have set limits.

Let's say you want to find the area under a curve. That's exactly what definite integrals do! In our original exercise, the definite integral \( \int_{0}^{1} 2 e^{x} \, dx \) is used to calculate this area between \( x = 0 \) and \( x = 1 \).

Here's how you can think about definite integrals:

• They calculate the net area: positive when above the x-axis, negative when below.
• The limits of integration, in this case 0 and 1, specify the interval on the x-axis.
• The calculation uses the antiderivative to evaluate the total change across this range.

Understanding how definite integrals work is essential for solving real-world problems in physics and engineering. They're a powerful tool to summarize changes over intervals.

Antiderivative

An antiderivative is the reverse of finding a derivative. Think of it as the opposite operation to differentiation.

Antiderivatives are known as the indefinite integrals of a function, from which definite integrals stem. In our problem, we were tasked with finding the antiderivative of \( 2e^{x} \).

Here’s what to remember about antiderivatives:

• The antiderivative of a basic exponential like \( e^{x} \) is itself because its derivative is \( e^{x} \) too.
• When you have a constant multiple, like 2 in \( 2e^{x} \), it remains constant while integrating.
• We compute definite integrals by evaluating the antiderivative at two points and calculating the difference.

Learning to find antiderivatives is foundational in mathematics. They bridge the gap between knowing the rate of change and understanding the cumulative effect or total amount.

Exponential Functions

Exponential functions are used frequently across different fields, symbolized by expressions like \( e^{x} \). They're known for their constant rate of growth or decay.

The original problem involved the function \( 2e^{x} \). Here’s why exponential functions are special:

• They're unique because their rate of change is proportional to their current value.
• The base of natural logarithms, \( e \), is approximately 2.718. It's the natural exponential.
• In this exercise, multiplying \( e^{x} \) by a constant factor, like 2, scales the function's value without changing its exponential nature.

Exponential functions are not just mathematical curiosities.
They're essential for modeling growth patterns in populations, radioactive decay, and even compound interest. When solving problems involving these functions, understanding their behavior and rules is key to mastering calculus.