Problem 7 - Calculus Early Transcendentals

Question

Use symmetry to evaluate the following integrals. $$\int_{-2}^{2} x^{9} d x$$

Step-by-Step Solution

Answer

Question: Determine the value of the definite integral ∫[-2, 2] x^9 dx using the property of symmetry and explain why it is so. Answer: The value of the definite integral ∫[-2, 2] x^9 dx is 0 because the function x^9 is odd, and the property of symmetry states that the integral of an odd function over symmetric bounds is equal to zero.

Step 1: Determine the nature of the function (odd or even)

Observe that the function given is \(f(x)=x^9\). We need to check if the function \(f\) has odd or even properties. For that, let's check for \(f(-x)\): $$f(-x)=(-x)^9=-x^9=-f(x)$$ As we can see, \(f(-x)=-f(x)\), which means that our function is an odd function.

Step 2: Applying symmetry property to odd functions

Since our function \(f(x)=x^9\) is odd, we can use the symmetry property for odd functions to evaluate the definite integral. The property states that the integral of an odd function over symmetric bounds is equal to zero, i.e., $$\int_{-a}^{a} f(x) dx = 0$$

Step 3: Evaluating the integral

Now, we can apply this property to the given integral to find its value: $$\int_{-2}^{2} x^9 dx = 0$$ The value of the definite integral is zero because the given function is odd and its domain is symmetric.

Key Concepts

Odd and Even Functions

When dealing with functions and their integrals, it's vital to determine if a function is odd or even. This distinction can reveal interesting properties about the function's behavior. Odd functions have the characteristic that for every input, the function's value at the negative of that input is the negative of its value at the original input. Mathematically, this is stated as:

• For an odd function, \( f(-x) = -f(x) \)

This means that odd functions are symmetric about the origin. In contrast, even functions satisfy the equality:

• For an even function, \( f(-x) = f(x) \)

These functions are symmetric about the y-axis. Identifying whether a function is odd or even is crucial in simplifying integrals, particularly when the limits of integration are symmetric.

Definite Integrals

Definite integrals provide the "net area" between a function and the x-axis over a closed interval. They are crucial in understanding many physical and geometric properties of functions. When calculating a definite integral, you find two values by evaluating the integral at the upper and lower limits of the interval, respectively, and then subtracting them. The integral is expressed as \( \int_{a}^{b} f(x) \, dx \), which calculates the area under the curve of \( f(x) \) from \( x = a \) to \( x = b \). Importantly, this area can be positive or negative, depending on whether the function is above or below the x-axis. This property of definite integrals perfectly complements the symmetry properties of odd and even functions. Recognizing symmetry can drastically simplify these calculations.

Symmetry Property

The symmetry property is a useful mathematical tool, especially in evaluating definite integrals. For odd functions, the symmetry property is particularly interesting. If an odd function \( f(x) \) is integrated over symmetric limits, say from \( -a \) to \( a \), the result is always zero:

• \( \int_{-a}^{a} f(x) \, dx = 0 \)

This occurs because the areas "above" and "below" the x-axis cancel each other out. To effectively use the symmetry property, it's important to first confirm if the function in question is odd.Even functions also have symmetric properties. For these, when integrated over a symmetric interval around the y-axis, the integration evaluation captures the doubled area under the curve from 0 to \( a \). This simplification can often lead to much easier calculations for functions that exhibit symmetrical properties.