Problem 405 - Calculus Volume 1

Question

Explain what is wrong with the following integral: $$\int_{1}^{2} \frac{d t}{\sqrt{1-t^{2}}}$$

Step-by-Step Solution

Answer

The integral is improper because the limits include values outside the domain of the integrand, making it undefined there.

Step 1: Identify the Function Inside the Integral

The given integral is \( \int_{1}^{2} \frac{dt}{\sqrt{1-t^{2}}} \). The function inside the integral is \( \frac{1}{\sqrt{1-t^{2}}} \).

Step 2: Recognize the Domain of the Integrand

The function \( \frac{1}{\sqrt{1-t^{2}}} \) has a domain where \( 1-t^2 > 0 \), or equivalently, \( t^2 < 1 \), meaning \( -1 < t < 1 \). Beyond this interval, the square root becomes undefined in the real number system.

Step 3: Check the Limits of Integration

The given limits of integration are from \( t = 1 \) to \( t = 2 \). The upper bound of the integral, \( t = 2 \), is outside the defined domain of \( -1 < t < 1 \).

Step 4: Identify the Issue with the Integral

Since the upper limit \( t = 2 \) is beyond the domain limit where \( \frac{1}{\sqrt{1-t^{2}}} \) is defined, the integral is improper as the integrand becomes undefined at \( t = 2 \). Also, the lower limit \( t = 1 \) is at the boundary of the domain, where the integrand tends towards infinity.

Step 5: Conclusion

The integral is improper because it attempts to evaluate values for which the integrand is not defined. This means the integral as it stands does not have a finite or meaningful value unless the limits are within the correct domain, and the integral is handled with limits.

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when dealing with any mathematical problem, especially integrals. The domain tells us the set of values for which a function is defined and can produce real number outputs. If we take the function within our integral, \( \frac{1}{\sqrt{1-t^{2}}} \), we need to ensure that the square root is valid in real numbers, meaning no negative values are under the square root.

Thus, for \( \sqrt{1-t^{2}} \), it is essential that \( 1-t^{2} > 0 \). Solving this gives the domain \(-1 < t < 1\). Any value of \( t \) outside this interval would produce a negative under the square root, resulting in an undefined function in the real number system. Recognizing this domain helps to prevent errors when setting limits of integration, avoiding undefined integrals.

Limits of Integration

Limits of integration denote the interval over which the function is to be integrated. They tell us where to start and stop our integration process. In the given integral, the limits were from \( t = 1 \) to \( t = 2 \). Here, an issue arises because these limits go beyond the defined domain of the function \( \frac{1}{\sqrt{1-t^{2}}} \), which is \(-1 < t < 1\).

When the limits of integration extend outside the domain, you encounter integration errors that can make the integral improper. An integral is considered improper if it attempts to evaluate points where the function is undefined or approaches infinity. Here, as \( t \) approaches 1 or exceeds it, the function becomes undefined. Therefore, always ensure that your limits are within the correct domain so the integral remains valid.

Undefined Integrands

An undefined integrand occurs when the function under the integral is not well-behaved across the limits of integration. In the integral \( \int_{1}^{2} \frac{dt}{\sqrt{1-t^{2}}} \), both limits cause issues leading to undefined conditions. The function \( \frac{1}{\sqrt{1-t^{2}}} \) becomes undefined at \( t = 2 \) because you end up with a negative inside the square root, and undefined at \( t = 1 \) because the function approaches division by zero, leading to a vertical asymptote.

When integrands are undefined at or between your integration limits, you cannot evaluate the integral directly and meaningfully. This is why adjustments using limit processes are necessary for proper evaluation, or by modifying the integral to ensure all values remain within the domain of definition. Remember, dealing with undefined integrands often requires special techniques like limit evaluation, ensuring all points are adequately managed.