Problem 16 - Applied Calculus

Question

Without calculation, what can you say about the relationship between the values of the two integrals: $$\int_{0}^{2} e^{x^{2}} d x \text { and } \int_{0}^{2} e^{t^{2}} d t ?$$

Step-by-Step Solution

Answer

The values of the two integrals are equal, as the variable is only a placeholder.

Step 1: Identify the Integral Expressions

The problem provides two integral expressions: \( \int_{0}^{2} e^{x^{2}} dx \) and \( \int_{0}^{2} e^{t^{2}} dt \). These are improper integrals of the same function but with different differential variables.

Step 2: Recognize the Role of Variables

The variable used in the integral (\(x\) vs. \(t\)) is a "dummy variable." This means that the variable is simply a placeholder, and does not affect the value of the integral itself.

Step 3: Establish the Equivalence

Since both integrals \( \int_{0}^{2} e^{x^{2}} dx \) and \( \int_{0}^{2} e^{t^{2}} dt \) have the same limits of integration and integrand, they are equivalent. The variable changes from \(x\) to \(t\) but the integral's value remains the same.

Key Concepts

Improper Integrals

Improper integrals come into play when the limits of integration are infinite or when the integrand becomes unbounded within the interval of integration. Initially, in the given exercise, we see definite integrals with finite limits, so they might not seem "improper" at first glance. Improper integrals can be tricky as they require us to consider these potential infinities or discontinuities in integration. We often handle them using limits, allowing us to approach these infinite values or undefined points. However, in this exercise, it's quite straightforward since both integrands are continuous and well-defined over the interval.

While this concept may not directly apply here, it's essential to recognize that improper integrals broaden the scope of integrable functions by reconsidering typical limitations.

Dummy Variables

In mathematics, particularly in calculus, we frequently encounter the concept of dummy variables. These are placeholders used within mathematical expressions or equations to generalize behaviors. In the context of integrals, dummy variables are especially common.
For instance, in the integrals \( \int_{0}^{2} e^{x^{2}} \, dx \) and \( \int_{0}^{2} e^{t^{2}} \, dt \), both \(x\) and \(t\) are dummy variables. They serve as symbols representing the variable of integration but don't hold specific values themselves. The integral's solution is independent of whether \(x\) or \(t\) is used. This is why you can switch them out without changing the integral's value.

Dummy variables aid in simplifying mathematics by allowing us to focus more on the structure and outcome of expressions rather than specific notation. Understanding their role helps in recognizing the symmetry and uniformity within mathematical operations.

Equivalence of Integrals

The concept of equivalence in integrals is pivotal when dealing with different notations or variables within the same integral form. In the provided exercise, we assess two integrals: \( \int_{0}^{2} e^{x^{2}} \, dx \) and \( \int_{0}^{2} e^{t^{2}} \, dt \). Although the variable changes from \(x\) to \(t\), the integrals are intrinsically identical. This equivalence stems from having the same integrand \(e^{u^2}\) where \(u\) might be \(x\) or \(t\) and the same limits of integration.
The core idea is recognizing that, if the only change in two integrals is the variable's name, then they are equal—a testament to the role of dummy variables.

Understanding this equivalence allows students and mathematicians alike to interchange variables for convenience, especially in complex numerical or symbolic manipulations, while maintaining the integrity and outcome of the integral values.