Problem 69 - Applied Calculus

Question

True or False: If \(f(-x)=f(x)\) for every \(x\) and if \(\int_{0}^{\infty} f(x) d x=7,\) then \(\int_{-\infty}^{0} f(x) d x=7\).

Step-by-Step Solution

Answer

True; symmetry of even functions leads to equal integrals over symmetric intervals.

Step 1: Identify the property of the function

We are given the function property that \(f(-x) = f(x)\) for every \(x\). This property indicates that the function \(f(x)\) is an even function. Even functions are symmetric about the y-axis, meaning their graph on one side of the y-axis is a mirror image of the other side.

Step 2: Understand the integral over a symmetric interval

For even functions, because of symmetry, the area under the curve from \(-a\) to \(0\) is equal to the area from \(0\) to \(a\). Hence, the integral \(\int_{-\infty}^{0} f(x) \, dx\) is equivalent to the integral \(\int_{0}^{\infty} f(x) \, dx\).

Step 3: Apply given value of integral

According to the problem, we know \(\int_{0}^{\infty} f(x) \, dx = 7\). Since \(f(x)\) is an even function, it follows that \(\int_{-\infty}^{0} f(x) \, dx\) should take the same value due to the symmetry property of even functions.

Step 4: Conclude based on symmetry

Thus, since the function is even and the integral from \(0\) to \(\infty\) is 7, the integral from \(-\infty\) to \(0\) must also be 7. The statement given in the exercise is true.

Key Concepts

Symmetry in Mathematics

The concept of symmetry in mathematics plays a crucial role, especially when dealing with functions. A function is said to be symmetric if one half of its graph is a mirror image of the other.

There are different types of symmetries, but in this context, we focus on even functions, which exhibit symmetry about the y-axis.

For a function to be even, it must satisfy the condition: - \( f(-x) = f(x) \) for all values of \( x \).

This means that if you were to fold the graph of the function along the y-axis, both halves would align perfectly. This symmetry implies that the behavior of the function from \( 0 \) to any positive value mirrors that from \( 0 \) to the corresponding negative value.

Properties of Integrals

Integrals are a fundamental concept in calculus, offering profound insights into functions’ behavior. When a function is even, the properties of integrals become particularly useful.

For even functions, these integral properties are greatly simplified:

• The integral over a symmetric interval, like from \(-a\) to \(a\), captures this symmetry beautifully.
• Because the function \( f(x) \) mirrors itself across the y-axis, the area under the curve from \(-a\) to \(0\) equals the area from \(0\) to \(a\).

The important takeaway is that if you know the integral of an even function from \(0\) to infinity, the integral from negative infinity to \(0\) will be identical. This is a direct result of the symmetry properties even functions satisfy.

Definite Integrals

Definite integrals are a tool that calculates the net area between the graph of a function and the x-axis over a specific interval. For an even function, definite integrals possess wonderfully predictable properties.

In the case of even functions, calculating the definite integral from \(-\infty\) to \(0\) produces the same result as from \(0\) to \(\infty\).
This is possible only because of the y-axis symmetry, central to even functions.

The process for computing a definite integral involves:

• Identifying the limits over which the integration occurs.
• Integrating the function within those bounds.
• Evaluating the result, which gives the exact area under the curve.

Understanding this symmetry simplifies working with definite integrals, allowing predictions about one part of the integral based on the other, saving time and effort in calculations.