Understanding Problems Based on the Property f(2a-x) = f(x)

When dealing with definite integrals in mathematics, certain properties of functions can greatly simplify the process of integration. One such property is the symmetry property, denoted as f(2a-x) = f(x). This property indicates that the function f(x) is symmetric about the line x = a. This symmetry can be exploited to solve definite integrals more efficiently.

Table of Differences and Important Points

Feature	f(2a-x) = f(x)	General Function
Symmetry	Symmetric about x = a	May not have symmetry
Integral Limits	Can be simplified when integrating from 0 to 2a	General limits apply
Function Behavior	f(a + h) = f(a - h) for any h	No such behavior guaranteed
Application	Simplifies integration problems	General integration methods needed

Formulas

The property f(2a-x) = f(x) can be used to simplify definite integrals of the form:

$$ \int_{0}^{2a} f(x) \, dx $$

By exploiting the symmetry, we can rewrite the integral as:

$$ \int_{0}^{2a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx $$

This is because the area under the curve from 0 to a is equal to the area from a to 2a due to the symmetry of the function about x = a.

Examples

Let's explore some examples to understand how to apply the property f(2a-x) = f(x) in solving definite integrals.

Example 1: Even Function

Consider the function f(x) = cos(x), which is an even function. This means that f(x) = f(-x). If we choose a = π/2, then f(π - x) = cos(π - x) = cos(x) = f(x).

Now, let's solve the integral:

$$ \int_{0}^{\pi} \cos(x) \, dx $$

Using the property, we can simplify this to:

$$ \int_{0}^{\pi} \cos(x) \, dx = 2 \int_{0}^{\frac{\pi}{2}} \cos(x) \, dx $$

Which can be easily evaluated to:

$$ 2 \left[ \sin(x) \right]_{0}^{\frac{\pi}{2}} = 2(1 - 0) = 2 $$

Example 2: Specific Function with Symmetry

Let's take a function f(x) = e^{(a-x)^2}, and we want to integrate it from 0 to 2a.

$$ \int_{0}^{2a} e^{(a-x)^2} \, dx $$

Since f(2a-x) = e^{(a-(2a-x))^2} = e^{(a-x)^2} = f(x), we can apply the property:

$$ \int_{0}^{2a} e^{(a-x)^2} \, dx = 2 \int_{0}^{a} e^{(a-x)^2} \, dx $$

This integral might still be challenging to solve, but the property has simplified the limits of integration.

Example 3: Application in Definite Integrals

Suppose we have the function f(x) = sin^2(x) and we want to integrate it from 0 to π.

$$ \int_{0}^{\pi} \sin^2(x) \, dx $$

Choosing a = π/2, we see that f(π - x) = sin^2(π - x) = sin^2(x) = f(x).

Applying the property:

$$ \int_{0}^{\pi} \sin^2(x) \, dx = 2 \int_{0}^{\frac{\pi}{2}} \sin^2(x) \, dx $$

This integral can be evaluated using a power-reduction formula:

$$ 2 \int_{0}^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx = \left[ x - \frac{\sin(2x)}{2} \right]_{0}^{\frac{\pi}{2}} = \frac{\pi}{2} $$

Conclusion

The property f(2a-x) = f(x) is a powerful tool in solving definite integrals, especially when the function exhibits symmetry about a certain point. By recognizing this property, one can often reduce the complexity of an integral and find the solution more efficiently. It is important to note that this property is not universal and can only be applied when the function in question satisfies the symmetry condition.