Problem 25 - Applied Calculus

Question

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{2} 2^{x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to approximately 2.885.

Step 1: Recognize the Integral Form

The given integral is \( \int_{1}^{2} 2^{x} \, dx \). This is an exponential function of the form \( a^x \), where \( a = 2 \), which cannot be integrated using basic algebraic methods. We will need to use a rule or a calculator/computer to find the numerical value.

Step 2: Recall the Integral Rule for Exponential Functions

The integral of an exponential function \( a^x \) is \( \frac{a^x}{\ln(a)} + C \), where \( C \) is the constant of integration. For the definite integral from 1 to 2, we will apply the limits.

Step 3: Apply the Limits to the Antiderivative

Substitute the limits into the antiderivative formula: \[ \int_{1}^{2} 2^{x} \, dx = \left[ \frac{2^{x}}{\ln(2)} \right]_{1}^{2} \]Calculate this to get: \[ \frac{2^{2}}{\ln(2)} - \frac{2^{1}}{\ln(2)} \]

Step 4: Evaluate the Expression

Calculate the expression: \[ \frac{4}{\ln(2)} - \frac{2}{\ln(2)} = \frac{4 - 2}{\ln(2)} = \frac{2}{\ln(2)} \].Using a calculator, find \( \ln(2) \approx 0.6931 \), and then \[ \frac{2}{0.6931} \approx 2.885 \].

Key Concepts

Exponential Functions

Exponential functions are an essential part of mathematics, appearing frequently in various real-world phenomena such as population growth, radioactive decay, and compound interest. At its core, an exponential function is expressed as \( a^x \), where \( a \) is a constant base and \( x \) is the variable exponent. This type of function grows (or decays) rapidly because the variable is in the exponent position. Unlike linear functions, exponential functions increase multiplicatively.
A key characteristic of exponential functions is that if the base \( a \) is greater than 1, such as in \( 2^x \), the function will demonstrate exponential growth. In contrast, if the base is between 0 and 1, the function will indicate exponential decay. These properties make exponential functions very powerful for modeling dynamic systems that change quickly over time.

Integration Rules

Integration rules are fundamental in calculus, providing techniques to find antiderivatives or integrals of functions. When dealing with exponential functions such as \( a^x \), there is a specific rule designed to help. The integral of \( a^x \) can be calculated using the formula:

• \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \)

This formula is crucial, as exponential functions often cannot be integrated using basic algebraic techniques.
In a definite integral like \( \int_{1}^{2} 2^{x} \, dx \), we apply the limits to the antiderivative to obtain the numerical value. Substitute the limits as follows:

• \( \frac{2^{2}}{\ln(2)} - \frac{2^{1}}{\ln(2)} \)

In this computation, we apply the integral rule specifically to tackle the increment within specified boundaries, effectively capturing the accumulated area under the curve.

Numerical Evaluation

Numerical evaluation refers to the process of obtaining a specific numerical result from an expression or equation, often via a calculator or computational software. This is especially useful for evaluating definite integrals that yield complex results or involve numbers like ln(2), which do not simplify neatly. After applying integration rules to find \( \int_{1}^{2} 2^{x} \, dx \), we reach an expression:

• \( \frac{2}{\ln(2)} \)

Here, \( \ln(2) \) is a constant approximately equal to 0.6931. Calculators can efficiently approximate this integral to be approximately 2.885, giving us the net area under the curve from \( x = 1 \) to \( x = 2 \).
This method of numerical evaluation allows us to handle complex integrals in practical scenarios, making it indispensable in both academic and real-world applications.