Problem 46 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove (by a substitution) that $$ \int_{a}^{b} f(-x) d x=\int_{-b}^{-a} f(x) d x $$

Step-by-Step Solution

Answer

By substituting \( u = -x \) and adjusting the limits, the integrals are shown to be equal.

Step 1: Understand the Problem

We need to show using substitution that the integral of \( f(-x) \) evaluated from \( a \) to \( b \) is equal to the integral of \( f(x) \) from \( -b \) to \( -a \).

Step 2: Apply the Substitution

Let's substitute \( u = -x \). Then \( du = -dx \) or \( dx = -du \).

Step 3: Change Limits of Integration

When \( x = a \), \( u = -a \) and when \( x = b \), \( u = -b \). This changes our integration limits from \( a \) to \( b \) to \( -a \) to \( -b \).

Step 4: Substitute into the Integral

Substituting \( u = -x \) and \( dx = -du \) into the expression: \[ \int_{a}^{b} f(-x) \,dx = \int_{-a}^{-b} f(u)(-du) = -\int_{-a}^{-b} f(u) \,du \]

Step 5: Reverse the Limits

Reversing the limits of integration from \( -a \) to \( -b \) to \( -b \) to \( -a \) gives us:\[ -\int_{-a}^{-b} f(u) \,du = \int_{-b}^{-a} f(u) \,du \]

Step 6: Rewrite in Terms of Original Function

Since \( u = -x \), \( f(u) = f(x) \). Therefore the integral is: \[ \int_{-b}^{-a} f(x) \,dx \] Thus, we have shown that \[ \int_{a}^{b} f(-x) \,dx = \int_{-b}^{-a} f(x) \,dx \]

Key Concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus. They allow us to find the exact accumulation of quantities. The definite integral of a function \( f(x) \) from \( a \) to \( b \) is denoted as \( \int_{a}^{b} f(x) \, dx \). This integral represents the area under the curve of \( f(x) \) between the limits \( a \) and \( b \).
Unlike indefinite integrals, definite integrals are evaluated over a specific interval and result in a numeric value, not a function.

• The limits \( a \) and \( b \) define the range over which the integral is calculated.
• The result is a fixed number representing a total quantity, such as area, volume, or total accumulated change.

An important property of definite integrals is that they can be split or altered with substitutions to simplify calculation without changing the result.

Integration by Substitution

Integration by substitution, often called u-substitution, is a technique used to simplify integrals. It's similar to the reverse of the chain rule in differentiation. The key idea is to substitute part of the integral with a new variable, typically \( u \), which makes the integral easier to evaluate.
This method involves a few steps:

• Identify the substitution: Choose a substitution that simplifies the integral. For example, in the substitution \( u = -x \), we transform the function within the integral.
• Change the differentials: Compute the differential \( du \) in terms of the original variable \( dx \). In our case, \( du = -dx \).
• Replace and integrate: Substitute \( u \) and \( du \) and then perform the integration with respect to \( u \).

Integration by substitution facilitates solving integrals that are complex or otherwise difficult to integrate directly.

Change of Variables

Change of variables is an integral part of the substitution process. It allows the transformation of the integration limits and integrand to a simpler or more convenient form. In the original exercise, the change of variable is performed by setting \( u = -x \).
Using the change of variables:

• New integration limits: When you substitute \( u = -x \), you must also adjust the limits of integration. The original limits \( a \) and \( b \) become \( -a \) and \( -b \) respectively.
• Integral simplification: Substitution often simplifies the integrand to a more manageable form, as seen when converting from \( f(-x) \) to \( f(u) \).

These transformations can greatly simplify the integration process, providing a clearer path to the solution.

Limits of Integration

The limits of integration denote the range over which a definite integral is calculated, listed as the lower and upper bounds. In substitution, limits of integration must match the variable in use.
Here's how this works in practice:

• When performing a substitution like \( u = -x \), the limits of integration also shift. If \( x \) ranged from \( a \) to \( b \), \( u \) now ranges from \( -a \) to \( -b \).
• Swapping the limits when necessary: If after substitution the limits are in descending order, reversing them will adjust the integral correctly, thus flipping its sign.

Correctly handling limits is vital, as failing to do so can lead to incorrect results. Always recalculate limits based on the new variable and make sure to revise their order when necessary.