Problem 26 - Calculus: Early Transcendentals

Question

Evaluate the integral. \( \displaystyle \int^5_{-5} e \,dx \)

Step-by-Step Solution

Answer

The integral evaluates to \( 10e \).

Step 1: Identify the constant integrand

Notice that the integrand, which is the function we are integrating, is a constant function, specifically, it is the constant 'e'. In integral notation, this means we are integrating the constant value 'e' with respect to x over a given interval.

Step 2: Recall the rule for integrating a constant

The rule for integrating a constant function is straightforward. For any constant 'c', the integral of c with respect to x over an interval \([a, b]\) is given by \(c imes (b - a)\).

Step 3: Substitute the constant and limits into the rule

Using the formula from Step 2, substitute 'e' for c, and -5 for 'a' and 5 for 'b': \[ ext{Integral} = e imes (5 - (-5)) \]

Step 4: Simplify the expression

Calculate the interval length first by computing the difference, \(5 - (-5) = 10\). Thus, \[ ext{Integral} = e imes 10 \]

Step 5: Conclude the solution

The integral is simply \[ 10e \]. Therefore, the evaluation of the integral \( \int_{-5}^{5} e \, dx \) gives \( 10e \).

Key Concepts

Constant Function

A constant function is one of the simplest types of functions in mathematics. It is defined by a single value that does not change, regardless of the input. For example,

• In the expression \( f(x) = c \), where \( c \) is a constant, \( f(x) \) always returns the value \( c \), no matter what value \( x \) takes.
• This is why it's called a "constant function" - the output is constant!

When dealing with integrals, a constant function makes the process straightforward because the function itself does not depend on any variable. Whether you're integrating a function like \( f(x) = e \), \( f(x) = 3 \), or any other constant value, the function's value remains unchanged throughout the interval that's being considered.

Definite Integral

A definite integral refers to the evaluation of an integral over a specific interval. The key elements of a definite integral include the integrand, the variable of integration, and the limits of integration. Let’s break it down:

• Integrand: In our example, the integrand is the constant \( e \).
• Variable of Integration: The variable of integration is \( x \), indicating we are considering changes with respect to \( x \).
• Limits of Integration: Here, we are looking at the interval from \( -5 \) to \( 5 \).

To find the definite integral of the constant \( e \), we calculate the area under the constant line represented by the function within these limits.This area is simply a rectangle's area whose height is the constant value (\( e \)) and whose width is the interval length (in this case, 10). Thus, the definite integral helps us find exact numerical values related to functions over defined regions.

Constant Integrand

A constant integrand means that the function we are integrating (the integrand) does not contain any variables. To alleviate any confusion:

• Integrand: This is the function being integrated, and in our case, it is just \( e \).

To integrate a constant function, we rely on a simple rule. When integrating a constant function like \( e \) over a range from \( a \) to \( b \), we multiply the constant by the width of the interval, which is \( b-a \).Thus,

• the integral of \( e \) over any interval \( [a, b] \) is the constant \( e \) times the length of the interval \( b - a \).

This makes integration straightforward, as you're calculating the area of a rectangle, where the height of the rectangle remains \( e \), constant throughout the interval. For our exercise, this results in \( 10e \) when evaluated from \( -5 \) to \( 5 \).