Problem 13 - Calculus Early Transcendentals

Question

Use symmetry to evaluate the following integrals. $$\int_{-10}^{10} \frac{x}{\sqrt{200-x^{2}}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the definite integral of the function \(f(x) = \frac{x}{\sqrt{200 - x^2}}\) from -10 to 10 using symmetry.

Step 1: Identify the integrand and the limits of integration

In this exercise, the integrand is given by the function \(f(x) = \frac{x}{\sqrt{200 - x^2}}\) and the limits of integration are \(a = -10\) and \(b = 10\). #Step 2: Check for symmetry#

Step 2: Check for symmetry

Since we need to use symmetry to evaluate this integral, let's check if the integrand is even, odd or has no symmetry: $$f(-x) = \frac{-x}{\sqrt{200 - (-x)^2}} = -\frac{x}{\sqrt{200 - x^2}} = -f(x)$$ As \(f(-x) = -f(x)\), the integrand is an odd function. #Step 3: Check the limits of integration for symmetry#

Step 3: Check the limits of integration for symmetry

In this case, the limits of integration are \(a = -10\) and \(b = 10\). Since \(a = -b\), the limits of integration are symmetric about the y-axis. #Step 4: Use symmetry to evaluate the integral#

Step 4: Use symmetry to evaluate the integral

Since the integrand is an odd function and the limits of integration are symmetric about the y-axis, we can use the property of odd functions and definite integrals to simplify the integral: $$\int_{-10}^{10} \frac{x}{\sqrt{200-x^{2}}} d x = 0$$ The definite integral of an odd function over symmetric limits is always equal to zero, so the final answer is: $$\int_{-10}^{10} \frac{x}{\sqrt{200-x^{2}}} d x = 0$$

Key Concepts

Odd Function Properties

An odd function is a type of function that has a specific symmetry with respect to the origin. Mathematically, a function f(x) is considered odd if it satisfies the condition that for every number x in the function's domain, it is true that f(-x) = -f(x). This means that if you plug in the negative of a number into an odd function, you'll get the opposite of what you'd get by plugging in the number itself.

One visual characteristic of odd functions is that their graphs are symmetric about the origin. This means if you rotate the graph 180 degrees around the origin, it will look identical. Odd functions often arise in real-world contexts where phenomena have this type of symmetry, such as certain wave patterns or forces in physics.

Symmetric Limits of Integration

Symmetric limits of integration refer to a situation in integral calculus where the lower and upper bounds of the definite integral are the same distance from the y-axis but on opposite sides. In other words, they are negative inverses of each other. For example, if the lower limit of integration is -a, then the upper limit would be +a, making the interval [-a, a] symmetric about the y-axis. This concept is especially valuable because it simplifies the evaluation of integrals of odd functions, as the area under the curve on one side of the y-axis will cancel out the area on the other side.

Definite Integrals

Definite integrals are a fundamental concept in calculus that quantify the accumulation of quantities or the net area under a curve on a graph between two points. They are represented as integrals with upper and lower limits and provide an exact value, unlike indefinite integrals which represent a family of functions. The process of finding the value of a definite integral is known as integration, and it involves determining the limit of a sum of areas of rectangles that approximate the area under a curve as their width approaches zero.

Definite integrals can be interpreted in various contexts, such as calculating distances, areas, volumes, and other physical quantities. They play a critical role in engineering, physics, statistics, and economics, where they are used to model and solve real-world problems.

Integral Symmetry Application

The concept of integral symmetry is a powerful tool when evaluating definite integrals. When confronted with an odd function and symmetric limits of integration, the integral will always equal zero. This is a direct result of the function's symmetry, as the areas above and below the x-axis on a graph cancel each other out. This principle of integral symmetry can dramatically simplify some integrals, sparing the need for complex calculations or numerical approximations.

Moreover, recognizing symmetry allows us to use the properties of definite integrals to make predictions about the behavior of functions and to understand the geometry of their graphs. This is an essential skill for students in mathematics and science fields, where symmetry often plays a critical role in the analysis of systems and phenomena.