Problem 36 - Thomas Calculus

Question

Evaluate the integrals. \begin{equation}\int_{-\ln 2}^{0} e^{-x} d x\end{equation}

Step-by-Step Solution

Answer

The value of the integral is 1.

Step 1: Identify the Integral

We need to evaluate the definite integral \( \int_{-\ln 2}^{0} e^{-x} \ dx \). This is a definite integral from \( x = -\ln 2 \) to \( x = 0 \).

Step 2: Finding the Antiderivative

The function to integrate is \( e^{-x} \). The antiderivative of \( e^{-x} \) is \( -e^{-x} \). Determining the antiderivative is the first step in solving the definite integral.

Step 3: Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, if \( F(x) \) is the antiderivative of \( f(x) \), then \( \int_{a}^{b} f(x) \ dx = F(b) - F(a) \). Using this theorem, we substitute the bounds into \( -e^{-x} \) to get \( F(0) - F(-\ln 2) \).

Step 4: Substitute and Simplify

Calculate \( F(0) \) and \( F(-\ln 2) \):- At \( x = 0 \), \( F(0) = -e^0 = -1 \).- At \( x = -\ln 2 \), \( F(-\ln 2) = -e^{-(-\ln 2)} = -e^{\ln 2} = -2 \).Now, substitute these into the formula: \( -1 - (-2) = -1 + 2 \).

Step 5: Calculate the Result

Simplify the expression: \( -1 + 2 = 1 \). This gives the value of the definite integral. Therefore, the integral evaluates to 1.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a critical concept that links the process of differentiation with integration. It forms the backbone of integral calculus and is often split into two main parts. The idea is simple: if you can find the antiderivative (or indefinite integral) of a function, you can solve the definite integral by applying this theorem.

The theorem states that if you have a continuous function, say \( f(x) \), on a closed interval \([a, b]\), and \( F(x) \) is an antiderivative of \( f(x) \), then:

• The definite integral of \( f(x) \) from \( a \) to \( b \) is given by \( F(b) - F(a) \).
• This means that you simply evaluate the antiderivative at the upper bound and subtract the evaluation at the lower bound.

Learners often marvel at how this simplifies evaluating definite integrals. You only need to focus on finding the antiderivative of the function, making calculations straightforward.

Antiderivative

An antiderivative of a function is another function whose derivative is the original function.

Let's say you have a function \( f(x) \). Its antiderivative, denoted often as \( F(x) \), satisfies \( F'(x) = f(x) \). This is crucial in solving more complex math problems.

In the context of definite integrals, once you find the antiderivative, as we did for \( e^{-x} \) with \( -e^{-x} \), you can effortlessly evaluate the integral using the Fundamental Theorem of Calculus.

• Finding the antiderivative is like reversing differentiation.
• For common functions like exponential functions, finding the antiderivative often follows familiar and standard rules.
• Once determined, the antiderivative allows us to use bounds to compute exact areas under curves on specified intervals.

Hence, mastering finding antiderivatives will make solving integrals much easier.

Exponential Functions

Exponential functions are essential in calculus due to their unique properties, particularly how they behave under differentiation and integration.

An exponential function can generally be expressed as \( e^{x} \) or \( e^{-x} \). These functions are distinct because they remain unchanged under differentiation. Their antiderivatives and derivatives are intuitive and predictable:

• The derivative of \( e^{x} \) remains \( e^{x} \).
• The derivative of \( e^{-x} \) results in \( -e^{-x} \).

This feature makes exponential functions particularly easy to integrate and differentiate.

For instance, when tackling the integration of \( e^{-x} \), recognizing its antiderivative as \( -e^{-x} \) allows leveraging the Fundamental Theorem of Calculus straightforwardly. Understanding these functions helps simplify calculus problems involving growth, decay, and continuous compounding scenarios.