Problem 60 - Essential Calculus

Question

True or False? In Exercises 55-60, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.. The value of \(\int_{2}^{2} \sin \left(x^{2}\right) d x\) is \(0 .\)

Step-by-Step Solution

Answer

True. The value of \(\int_{2}^{2} \sin \left(x^{2}\right) dx\) is indeed \(0\) because the integral over an interval of length zero is always zero regardless of the function integrated.

Step 1: Understand the problem

In this exercise, the given integral is \(\int_{2}^{2} \sin \left(x^{2}\right) d x\). This integral represents the signed area under the curve \(y=\sin(x^2)\) between \(x=2\) and \(x=2\), which is a single point. Therefore, the integral over an interval of length zero is always zero regardless of the function integrated.

Step 2: Evaluate the integral

Since the upper limit of the integral equals the lower limit, the integral evaluates to zero. Mathematically, an integral taken over a definite range where the upper limit equals to the lower limit will always be 0, regardless of the function. In other words, this is the property of definite integral: \(\int_{a}^{a} f(x) dx = 0\) for any function \(f(x)\).

Key Concepts

properties of definite integrals

Definite integrals are a powerful tool used to calculate the signed area under a curve over a specific interval on the x-axis. They provide valuable information about the total accumulation of a quantity, like distance or area. One key property of definite integrals is the linearity property, which says that you can split the integral over an interval

• If you have two functions, say, \( f(x) \) and \( g(x) \), then \( \int_{a}^{b} [f(x) + g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx \).
• For any constant \(c\), \( \int_{a}^{b} c \cdot f(x) \, dx = c \cdot \int_{a}^{b} f(x) \, dx \).

This allows you to handle more complex integrals by breaking them into simpler parts.

The restriction on upper and lower limits in the definite integrals has an important implication. If you have the same upper and lower limits, the integral will be zero, as we'll see in the next section.

zero interval integral

The zero interval integral is an interesting concept in mathematics. When you integrate a function over an interval where the limits of integration are the same, the integral always evaluates to zero.

• This is because there is no distance between the upper and lower limits for the function to "cover" or "summarize", so mathematically, nothing is accumulated.

For example, consider the integral \( \int_{2}^{2} \sin(x^2) \, dx \). Since the interval length is zero \((\text{2} - \text{2} = \text{0})\), the integral evaluates to zero regardless of the function \( \sin(x^2) \).

This concept is a specific property of definite integrals and showcases an inherent simplicity: zero length in an interval leads to zero accumulation of area.

evaluating integrals

Evaluating integrals effectively requires combining our understanding of their properties and applying them to specific problems. Whether you're trying to find the area under a curve, compare accumulated quantities, or solve more complex equations, integrals provide the framework.

Here are some basic steps you might follow when evaluating:

• Identify the limits of the integral to determine the interval over which you are evaluating. This helps in deciding the area or range you are focusing on.
• Consider the properties: Use linearity, sum, and any special properties like zero interval to simplify your calculations when possible.
• Calculate the integral using the fundamental theorem of calculus, if applicable, to find the antiderivative and evaluate at the endpoints of your interval.

The large scope of integrals in mathematics allows them to be used in diverse analyses, from physics to economics. Each step in the integration process can build a deeper understanding of how quantities change across specified intervals.