Problem 85 - Calculus Single Variable

Question

Let \(f:[-a, a] \rightarrow \mathbb{R}\) be continuous and even \((f(-x)=\) \(f(x)\) ). Show that $$ \int_{-a}^{a} f(x) d x=2 \int_{0}^{a} f(x) d x $$

Step-by-Step Solution

Answer

The integral of an even function over a symmetric interval is twice the integral from 0 to the endpoint.

Step 1: Understanding Even Functions

An even function is symmetric about the y-axis. This means that for every point \((x, f(x))\), there exists a point \((-x, f(x))\). Therefore, \(f(-x) = f(x)\) is true for all \(x\) within the domain.

Step 2: Analyzing the Integral of an Even Function

The integral of an even function over a symmetric interval \([-a, a]\) can be split into two symmetric parts: from \(-a\) to 0 and from 0 to \(a\). Thus, we have: \[\int_{-a}^{a} f(x) \, dx = \int_{-a}^{0} f(x) \, dx + \int_{0}^{a} f(x) \, dx\] due to the properties of definite integrals.

Step 3: Substituting the Definite Integral Bounds

Since \(f\) is even, \(\int_{-a}^{0} f(x) \, dx\) and \(\int_{0}^{a} f(x) \, dx\) have the same value. This is because for each \(x\) in \([0, a]\), the corresponding \(-x\) in \([-a, 0]\) satisfies \(f(-x) = f(x)\). Hence:\[\int_{-a}^{0} f(x) \, dx = \int_{0}^{a} f(x) \, dx\]

Step 4: Combining the Integrals

Combine the integrals from Step 3 to show that the original integral over \([-a, a]\) is double the integral from 0 to \(a\):\[\int_{-a}^{a} f(x) \, dx = \int_{0}^{a} f(x) \, dx + \int_{0}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx\] This equation confirms the given proposition.

Key Concepts

Symmetry in Calculus

Symmetry in calculus is a powerful concept that allows us to understand the behavior of functions through their symmetric properties. Specifically, even functions have symmetry about the y-axis. This means that for any input value \( x \), the output \( f(x) \) is the same as \( f(-x) \).

This property simplifies the calculation of definite integrals over symmetric intervals. For an interval running from

• \(-a\) to \(a\),
• where \(a\) is a positive real number,
• this symmetry suggests a balanced distribution of function values on either side of the y-axis.

If a function \( f(x) \) is even, you can effectively calculate its integral over the entire symmetric interval by focusing on the interval from \(0\) to \(a\), which simplifies the computation considerably.

Calculus symmetry hence provides an advantage by reducing the amount of work needed for integration while maintaining accuracy.

Definite Integrals

Definite integrals are a fundamental tool in calculus that allow us to calculate the net area under a curve over a specific interval. This area is referred to as the integral of the function over that interval.

• The notation \( \int_{a}^{b} f(x) \, dx \) represents the integral of \(f(x)\) from \(a\) to \(b\).
• This results in a numerical value that represents the accumulated areas between the graph of \(f(x)\) and the x-axis.

When dealing with definite integrals of even functions over symmetric limits

• such as from \(-a\) to \(a\),

a unique property emerges. Because of the symmetry, the area calculated over the negative side of the y-axis mirrors the area on the positive side. This means that the definite integral can be expressed as twice the integral from

• \(0\) to \(a\).

So, rather than evaluating \(\int_{-a}^{a} f(x)\, dx\) directly, you can compute \(2 \int_{0}^{a} f(x)\, dx\), leveraging the even nature of the function.

Properties of Even Functions

Even functions have distinct properties that highlight their symmetry about the y-axis. One defining characteristic is that an even function \(f(x)\) satisfies the condition \(f(-x) = f(x)\) for every \(x\) in its domain.
These properties are extremely valuable, especially in calculus, as they facilitate advanced operations like integration.

• An even function will always produce the same value for \(x\) and \(-x\).
• This repeating value pattern allows definite integrals over symmetric intervals \([-a, a]\) to be effectively simplified.
• Specifically, it supports the simplification to \(2 \int_{0}^{a} f(x)\, dx\), because the two halves of the interval contribute equally to the total integral.

The predictability of even functions means that they are extensively used in mathematical analysis and applications, providing elegant solutions and insights to complex calculus problems. Their symmetrical properties open the door to easier computations and clearer visual understandings of the behavior of functions.