Problem 48 - Calculus and its applications

Question

Evaluate. $$ \int_{0}^{2} e^{4 x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{4}(e^8 - 1) \).

Step 1: Understand the Type of Integral

This is a definite integral of the form \( \int e^{ax} \ dx \). Our goal is to find the antiderivative of the function \( e^{4x} \) and then evaluate it from 0 to 2.

Step 2: Find the Antiderivative

The antiderivative of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \) where \( C \) is the constant of integration. For \( e^{4x} \), the antiderivative is \( \frac{1}{4}e^{4x} + C \). Since we're evaluating a definite integral, we can ignore the constant \( C \).

Step 3: Evaluate at the Upper and Lower Limits

We now evaluate \( \frac{1}{4}e^{4x} \) from 0 to 2. First, plug in the upper limit: \( \frac{1}{4}e^{4 \cdot 2} = \frac{1}{4}e^8 \). Second, plug in the lower limit: \( \frac{1}{4}e^{4 \cdot 0} = \frac{1}{4} \).

Step 4: Calculate the Definite Integral

Subtract the evaluation at the lower limit from the evaluation at the upper limit: \( \frac{1}{4}e^8 - \frac{1}{4} = \frac{1}{4}(e^8 - 1) \). This is the value of the integral.

Key Concepts

Antiderivative

An antiderivative is a function whose derivative is the given function. When we talk about integrals, especially in calculus, the antiderivative plays a crucial role. To better explain, finding an antiderivative corresponds to "reversing" differentiation.
For example, when provided with a function like \( e^{4x} \), our task is to figure out which function, when differentiated, results in \( e^{4x} \).
The general rule for an antiderivative of \( e^{ax} \) is \( \frac{1}{a}e^{ax} + C \). Here, \( C \) stands for the constant of integration.

• The constant \( a \) is a coefficient in the exponent of \( e \).
• Your goal is to adjust the coefficient to obtain the correct antiderivative.

In the case of definite integrals, like the one in the exercise, we skip the constant \( C \) because we are computing an exact value over a specific interval.

Exponential Functions

Exponential functions are mathematical functions denoted by \( e^{x} \), where \( e \) represents Euler’s number—approximately 2.718—in mathematics. They appear in many natural and financial growth scenarios.
One key aspect of exponential functions is their rate of growth, a characteristic that sets them apart from polynomial or linear functions. They exhibit a constant proportional growth rate.

• They are defined for all real numbers and are always positive.
• The derivative and the antiderivative of an exponential function often involve the function itself, which showcases its naturally recursive property.

When calculating integrals or derivatives involving exponential functions, the rules are special because the function \( e^{ax} \) maintains its form but involves scaling factors like \( a \). Mastery of these functions is crucial for many areas of calculus.

Integral Calculus

Integral calculus deals with integrals and is used to determine quantities like areas, volumes, and total values of a function over an interval. A definite integral, such as \( \int_{0}^{2} e^{4x} dx \), evaluates the net area under a curve from a lower to an upper limit.
To solve a definite integral, follow these steps:

• Identify the function you wish to integrate, for instance, \( e^{4x} \).
• Find its antiderivative \( \frac{1}{4}e^{4x} \).
• Evaluate the antiderivative at the upper and lower limits.
• Subtract the value obtained at the lower limit from that at the upper limit.

This subtraction gives the net area or the overall value we seek. Integral calculus is powerful, especially when dealing with infinite series, calculating probabilities, or understanding physics and engineering problems.