Problem 138 - Calculus Volume 1

Question

If \( \int_{1}^{5} \sqrt{1+t^{4}} d t=41.7133 \ldots, \) what is \(\int_{1}^{5} \sqrt{1+u^{4}} d u ?\)

Step-by-Step Solution

Answer

The value is 41.7133...

Step 1: Understand the Problem

We are given the integral \( \int_{1}^{5} \sqrt{1+t^{4}} \, dt=41.7133 \ldots \) and are asked to find \( \int_{1}^{5} \sqrt{1+u^{4}} \, du \). Notice that the integral involves a substitution from \( t \) to \( u \), but the function inside the integral and the limits of integration remain exactly the same.

Step 2: Recognize the Constants of Integration

In definite integrals, the variable of integration (\( t \) or \( u \)) is a dummy variable. It simply represents the variable over which integration is performed. Thus, changing \( t \) to \( u \) does not change the value of the integral as long as the limits and the integrand stay the same.

Step 3: Solve the Integral

Recognizing that the integral of \( \int_{1}^{5} \sqrt{1+t^{4}} \, dt \) and \( \int_{1}^{5} \sqrt{1+u^{4}} \, du \) are the same since the function and limits are unchanged, the value of both integrals is equal.

Key Concepts

Substitution

When dealing with integrals, substitution is a technique used to simplify the integration process. By replacing the variable of integration with another variable, you can often make the integral easier to solve. In this exercise, although there appears to be a substitution from \( t \) to \( u \), no actual transformation of the integral is performed beyond renaming the variable.
This type of substitution does not involve changing the limits of integration, or the form of the function. Hence, the integrals \( \int_{1}^{5} \sqrt{1+t^{4}} \, dt \) and \( \int_{1}^{5} \sqrt{1+u^{4}} \, du \) remain identical.

• Substitution is often used to translate a complex integral into a simpler one.
• However, in this case, it shows the interchangeable nature of integration variables when limits and integrand do not change.

This underlines substitution's fundamental principle: the choice of variable is arbitrary as long as the function and limits remain constant.

Dummy Variable

The concept of a dummy variable in integrals can be confusing at first, but it’s critical to understand. In integration, the variable (such as \( t \) or \( u \)) is termed a "dummy variable." This means it simply serves as a placeholder to record the process of integration.
This is an important point because it highlights the fact that the letter used doesn't matter as long as it holds the same role.

• Think of dummy variables like x and y in equations; they are just symbols we use to express ideas.
• In our problem, \( t \) and \( u \) both serve as dummy placeholders with no real impact on the integral's value.

All that truly matters is the form of the function and the limits of integration. As long as these remain the same, the integral's value will not change, regardless of the letter used.

Limits of Integration

Limits of integration define the interval over which you're integrating a function. They are a key element in calculating a definite integral, as they specify the bounds within which you sum the area under the curve.
For definite integrals, even if you change the variable name from \( t \) to \( u \), the limits \( 1 \) and \( 5 \) remain unchanged.

• The limits ensure the integral results in a specific numerical value.
• Without changing these limits, the integral computes the same exact area, even with different variable names.

Given the identical limits and integrand in the original exercise, the outcome of both integrals \( \int_{1}^{5} \sqrt{1+t^{4}} \, dt \) and \( \int_{1}^{5} \sqrt{1+u^{4}} \, du \) remains equivalent. Limits solidify the result, marking the integration's start and end points.