Problem 51 - Calculus Early Transcendentals

Question

Suppose \(f\) is an even function and $$\int_{-8}^{8} f(x) d x=18$$. a. Evaluate \(\int_{0}^{8} f(x) d x\) b. Evaluate \(\int_{-8}^{8} x f(x) d x\)

Step-by-Step Solution

Answer

Answer: The integral of the given even function \(f(x)\) from \(0\) to \(8\) is 9, and the integral of the product of the given function and \(x\) from \(-8\) to \(8\) is 0.

Step 1: Review properties of even functions

Recall that for an even function \(f(x)\), we have \(f(-x) = f(x)\). This property results in the integral of an even function over a symmetric interval having a special property. More specifically, \(\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx\).

Step 2: Evaluate \(\int_{0}^{8} f(x) d x\)

We know that \(\int_{-8}^{8} f(x) d x=18\). Since \(f(x)\) is an even function, we can use the property mentioned in Step 1 to rewrite this as \(2 \int_{0}^{8} f(x) dx = 18\). Now, we can solve for \(\int_{0}^{8} f(x) d x\) by dividing both sides of the equation by 2:$$\int_{0}^{8} f(x) d x = \frac{18}{2} = 9$$. Thus, \(\int_{0}^{8} f(x) d x = 9\).

Step 3: Review properties of odd functions and even functions products

Recall that if we multiply an even function by an odd function, we get an odd function. Moreover, the integral of an odd function over a symmetric interval always equals zero. That is, \(\int_{-a}^{a} h(x) dx = 0\) for any odd function \(h(x)\).

Step 4: Evaluate \(\int_{-8}^{8} x f(x) d x\)

In this part, we want to evaluate \(\int_{-8}^{8} x f(x) d x\). Here, we can recognize that the function \(xf(x)\) is a product of an odd function (\(x\)) and an even function (\(f(x)\)). As mentioned in Step 3, this product results in an odd function. Therefore, the integral of \(x f(x)\) over the symmetric interval \([-8, 8]\) is zero:$$\int_{-8}^{8} x f(x) d x = 0$$.

Key Concepts

Integral Properties

The integral properties of functions reveal several interesting facets about how functions behave over particular intervals. One foundational element is the understanding of integrals themselves, which are essentially a measure of total accumulation, such as an area under a curve. Here are some key properties:

• Linearity: Integration respects addition and scalar multiplication. For functions \(f(x)\) and \(g(x)\) and scalars \(a\) and \(b\), \(\int (a f(x) + b g(x)) dx = a \int f(x) dx + b \int g(x) dx\).
• Additivity over intervals: The integral over a large interval can be broken down into smaller intervals: \(\int_{a}^{c} f(x) dx = \int_{a}^{b} f(x) dx + \int_{b}^{c} f(x) dx\).

Understanding these properties is crucial when working with specific types of functions such as even and odd functions over symmetric intervals.

Symmetric Intervals

When we integrate functions over symmetric intervals, from \(-a\) to \(a\), peculiar properties can arise. A key concept is how this symmetry impacts even and odd functions:

• If a function \(f(x)\) is even, meaning \(f(-x) = f(x)\), the integral over a symmetric interval can be simplified. This is because the areas on both sides of the y-axis are symmetrical. Thus, the integral can be expressed as double the integral from 0 to \(a\): \(\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx\).
• Symmetric intervals simplify calculations, especially for even functions, as in the given exercise where \(\int_{-8}^{8} f(x) dx = 18\) became \(2 \int_{0}^{8} f(x) dx = 18\).

These properties greatly simplify evaluating certain integrals and understanding the comprehensive behavior of the function over symmetric ranges.

Odd Function Integrals

The concept of odd functions adds another layer of intrigue when dealing with integrals. An odd function satisfies the condition \(f(-x) = -f(x)\). This property leads to some unique results:

• When integrating an odd function over a symmetric interval, the result is always zero. This is because the contributions from \(-a\) to 0 and from 0 to \(a\) cancel each other out completely: \(\int_{-a}^{a} f(x) dx = 0\).
• In the exercise context, the function \(x f(x)\) was noted to become odd since \(x\) itself is an odd function, and the product of an even function and an odd function is always odd. Consequently, \(\int_{-8}^{8} x f(x) dx = 0\), reflecting this integral property.

Recognizing whether a function is odd or not is essential in quickly determining the outcome of integrals over symmetric intervals and can streamline the problem-solving process significantly.