Question

Use symmetry to help you evaluate the given integral. $$ \int_{-1}^{1} x e^{-4 x^{2}} d x $$

Short answer

The integral is 0 because the integrand is an odd function.

Step-by-step solution

Identify the Symmetry

The given integral is symmetric about the y-axis because the limits of integration are from \(-1\) to \(1\) and the function is composed of even and odd functions. The function inside the integral, \(x e^{-4x^2}\), is odd because it can be written as \(f(-x) = -f(x)\).

Apply Symmetry Property

When integrating an odd function over a symmetric interval \([-a, a]\), the result is 0. This is due to the fact that the area under the curve on one side of the y-axis is equal and opposite to the area on the opposite side.

Key concepts

Odd Function

An odd function is a special type of function in calculus that has a specific symmetry property. Odd functions are defined by the relation \( f(-x) = -f(x) \). This means that if you input the negative of any value into the function, the result is the negative of what you would get by inputting the positive value.
Some key characteristics of odd functions include:

• Graphically, odd functions exhibit rotational symmetry about the origin. This means that if you rotate the entire graph 180 degrees around the origin, it looks unchanged.
• Many functions, such as \(x^3\), \(sin(x)\), and the function given in the exercise \(x e^{-4x^2}\), are odd.

Understanding odd functions is crucial when evaluating integrals over symmetric intervals, as their unique symmetry properties simplify calculations greatly.

Symmetry in Integrals

Symmetry can be a powerful tool when evaluating definite integrals. It often simplifies the computation, especially when dealing with functions defined on symmetric intervals like \([-a, a]\).
There are different types of symmetry associated with functions:

• If a function is odd and the interval is symmetric around the y-axis, the integral of this function over such interval results in 0. This occurs because the contributions to the integral from negative and positive portions of the domain cancel each other out.
• For even functions, which are symmetric about the y-axis (discussed later), integrals over symmetric bounds can often be simplified as well.

Recognizing a function's symmetry type is an essential skill. It provides insight into potential simplifications and reduces the need for lengthy calculations.

Definite Integral

A definite integral is a fundamental concept in calculus that provides a way to calculate the "net area" under a curve and between the x-axis over a specified interval. Specifically, the definite integral of a function \(f(x)\) from \(a\) to \(b\) is denoted as \(\int_a^b f(x) \, dx\).
In definite integrals:

• The limits of integration \(a\) and \(b\) define the interval over which the function is integrated.
• The value of the definite integral can represent physical quantities such as the total accumulated change or area under the curve.

In the given exercise, the integration from \(-1\) to \(1\) takes advantage of the symmetry of the function to evaluate the integral quickly.

Even Function

Even functions have a symmetry where \(f(x) = f(-x)\). This means that the graph of an even function is mirrored on the y-axis.
Some characteristics of even functions include:

• They have symmetry about the y-axis, meaning that the left and right sides of the graph are identical.
• Common examples include \(x^2\), \(cos(x)\), and constant functions.

When integrating even functions over symmetric intervals, it's often possible to use their symmetry to simplify calculations. For instance, instead of integrating over \([-a, a]\), the area can be calculated over \([0, a]\) and then doubled due to the symmetry about the y-axis.