Question

How does \(\int_{-b}^{-a} f(x) d x\) compare with \(\int_{a}^{b} f(x) d x\) when \(f\) is an even function? An odd function?

Short answer

If \(f\) is even, integrals are equal; if odd, they cancel out.

Step-by-step solution

Understanding Even and Odd Functions

A function is even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \). It is odd if \( f(-x) = -f(x) \) for all \( x \). We will use these properties to compare the integrals.

Setting Up the Integrals for Even and Odd Functions

We need to compare \(\int_{-b}^{-a} f(x) \, dx\) with \(\int_{a}^{b} f(x) \, dx\). We will evaluate each integral type based on the property of \(f(x)\) being even or odd.

Evaluating the Integral for an Even Function

For an even function, \(f(-x) = f(x)\). Thus, \(\int_{-b}^{-a} f(x) \, dx = \int_{b}^{a} f(-t) \, d(-t) = \int_{b}^{a} f(t) \, dt\). Changing bounds gives \(-\int_{a}^{b} f(t) \, dt\), which after reversing is \(\int_{a}^{b} f(t) \, dt\). Hence, both integrals are equal.

Evaluating the Integral for an Odd Function

For an odd function, \(f(-x) = -f(x)\). Thus, \(\int_{-b}^{-a} f(x) \, dx = \int_{b}^{a} f(-t) \, d(-t) = \int_{b}^{a} -f(t) \, dt\) after changing variables, which simplifies to \(-\int_{a}^{b} f(t) \, dt\). These integrals are opposites, meaning they sum to zero: \(\int_{a}^{b} f(x) \, dx = -\int_{-b}^{-a} f(x) \, dx\).

Concluding the Comparison

For an even function, the integrals \(\int_{-b}^{-a} f(x) \, dx\) and \(\int_{a}^{b} f(x) \, dx\) are equal. For an odd function, the integrals have equal magnitude but opposite signs, meaning their sum is zero.

Key concepts

Even Function

An even function is one of the simpler types of functions, defined by the property that for every input, its result is the same whether you use the positive or negative version of that input. In mathematical terms, an even function fulfills the condition \( f(-x) = f(x) \) for all \( x \). This reflects perfectly symmetrical behavior around the y-axis.

• Think of common examples like \( f(x) = x^2 \) or \( f(x) = \cos x \), which look mirrored when folded over the y-axis.
• The symmetry of even functions means they handle integrals from negative to positive limits in a straightforward manner.
• When you evaluate an integral of an even function over symmetric limits, such as \( \int_{-b}^{b} f(x) \, dx \), it is equivalent to twice the integral from 0 to \( b \). This is because the area on both sides of the y-axis is identical.

Odd Function

Odd functions introduce a different kind of symmetry related to rotation. Specifically, an odd function satisfies \( f(-x) = -f(x) \). This means that if you rotate the graph 180 degrees around the origin, it looks the same.

• Common examples include \( f(x) = x^3 \) or \( f(x) = \sin x \), both of which exhibit this rotational symmetry.
• For integrals, this property has an interesting effect. Over symmetric limits like \( (-b, b) \), the positive and negative parts cancel each other out.
• As a result, the definite integral of an odd function over such a symmetric interval is zero: \( \int_{-b}^{b} f(x) \, dx = 0 \).

Definite Integrals

Definite integrals form the backbone of analysis in calculus, providing a precise measure of the area under a curve between two specific points. When working with definite integrals, it's crucial to set the limits correctly, as this determines the segment of the function you are analyzing.

• The notation \( \int_{a}^{b} f(x) \, dx \) signifies integrating \( f(x) \) from \( x = a \) to \( x = b \).
• By integrating over an interval, you're effectively summing up infinitesimally small rectangles under the curve, giving you a total area value.
• This concept is vital in comparing integrals for even and odd functions as it determines whether the contributions from each side of the y-axis add up (in even functions) or cancel out (in odd functions).

Function Properties

Understanding the properties of functions is essential in calculus as it determines how they behave under various mathematical operations, such as integration. Properties like symmetry, periodicity, and boundedness can drastically affect the outcome of integrals and other calculations.

• Even and odd functions, characterized by their respective symmetrical properties, provide clear examples of how these properties influence integration results.
• Recognizing whether a function is even or odd can simplify the calculation of definite integrals, as seen in the provided examples: even functions make integral comparison easier, while odd functions often lead to integral cancellation over symmetric limits.
• Awareness of function properties also helps in predicting the behavior of complex integrals, especially when dealing with large intervals or variable transformations.