Problem 31 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(29-50.\) $$\int_{\ln 2}^{\ln 3} e^{x} d x$$

Step-by-Step Solution

Answer

The integral evaluates to 1.

Step 1: Identify the Integral

We need to evaluate the definite integral \( \int_{\ln 2}^{\ln 3} e^x \, dx \). This means we need to find the area under the curve \( e^x \) from \( x = \ln 2 \) to \( x = \ln 3 \).

Step 2: Antiderivative of the Integrand

Recognize that the derivative of \( e^x \) is \( e^x \). Therefore, the antiderivative of \( e^x \) is \( e^x \) itself.

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, we evaluate the antiderivative at the bounds: \( [e^x]_{\ln 2}^{\ln 3} = e^{\ln 3} - e^{\ln 2} \).

Step 4: Simplify the Expression

Calculate \( e^{\ln 3} \) and \( e^{\ln 2} \). Recall that \( e^{\ln a} = a \), so \( e^{\ln 3} = 3 \) and \( e^{\ln 2} = 2 \).

Step 5: Compute the Definite Integral

Subtract the lower bound result from the upper bound result: \( 3 - 2 = 1 \).

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a central idea in calculus that connects the concept of differentiation with integration. It essentially tells us that integration is the reverse process of differentiation. Here's how it works in the context of definite integrals:

• The theorem states that if you have a continuous function that you can find an antiderivative for, you can use it to evaluate definite integrals.
• Specifically, the value of the definite integral of a function from point A to point B is the difference between the values of any of its antiderivatives evaluated at these two points.

This means that for \(\int_a^b f(x) \, dx\), you find an antiderivative F of f, and then compute \(F(b) - F(a)\). This tight connection between integrals and derivatives helps calculate areas and accumulations effectively.It is both a powerful theoretical tool and a practical technique for evaluating integrals.

Antiderivative

An antiderivative, in simple terms, is a function whose derivative is the original function you started with. When you find an antiderivative, you're essentially looking for a function F(x) whose differentiation results in the given function f(x). Let's look at key points about antiderivatives:

• Antiderivatives are also referred to as indefinite integrals.
• Finding the antiderivative is crucial for solving definite integrals using the Fundamental Theorem of Calculus.

Take the function \(e^x\): The derivative of \(e^x\) is \(e^x\), so naturally, its antiderivative is itself. Understanding antiderivatives brings balance between differentiation and integration, two cornerstone concepts of calculus.

Exponential Functions

Exponential functions are a type of mathematical function that features a constant base raised to a variable exponent. They have the form \(a^x\), where \(a\) is a positive constant. One of the most common and important exponential functions is the natural exponential function, represented as \(e^x\).

• The number \(e\) is a special constant approximately equal to 2.71828.
• The natural exponential function has unique properties, especially when it comes to derivatives and antiderivatives.

The function \(e^x\) is particularly interesting because it is its own derivative and antiderivative, making it incredibly useful in calculus.Exponential functions model growth and decay processes, which are common in natural and social sciences.Their behavior can elucidate a wide range of phenomena, from population growth to radioactive decay.