Problem 9 - Calculus Concepts

Question

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{10}^{\infty} 3 x^{-2} d x $$

Step-by-Step Solution

Answer

The value of the improper integral is \(\frac{3}{10}\).

Step 1: Set Up the Improper Integral

To evaluate the improper integral \( \int_{10}^{\infty} 3x^{-2} \, dx \), we begin by setting up a limit that replaces the infinity symbol. Thus, the integral becomes \( \lim_{b \to \infty} \int_{10}^{b} 3x^{-2} \, dx \).

Step 2: Integrate the Function

Now, we integrate the function \(3x^{-2}\). The antiderivative of \(x^{-2}\) is \(-x^{-1}\). Therefore, the antiderivative for \(3x^{-2}\) is \(-3x^{-1}\). So, \(\int 3x^{-2} \, dx = -\frac{3}{x} + C\), where \(C\) is the constant of integration.

Step 3: Evaluate the Definite Integral

Using the antiderivative, we have \( \int_{10}^{b} 3x^{-2} \, dx = \left[-\frac{3}{x}\right]_{10}^{b} \). This evaluates to \[-\frac{3}{b} - \left(-\frac{3}{10}\right) = -\frac{3}{b} + \frac{3}{10}.\]

Step 4: Take the Limit as b Approaches Infinity

Finally, we take the limit of the expression as \(b \to \infty\): \[ \lim_{b \to \infty} \left(-\frac{3}{b} + \frac{3}{10} \right). \] As \(b\) approaches infinity, \(-\frac{3}{b}\) approaches 0. Therefore, the limit simplifies to \[0 + \frac{3}{10} = \frac{3}{10}.\]

Key Concepts

Definite Integration

Definite integration is a fundamental concept in calculus that deals with the calculation of the integral of a function over a specific interval. In simpler terms, it helps us to find the total accumulation, or area under a curve, between two points. In the exercise provided, we are interested in the integral of the function \(3x^{-2}\) from 10 to infinity. Since one of the limits is infinity, the definite integral becomes improper, and we must utilize limits to manage this.

• The process of definite integration involves two boundaries, a lower and an upper limit, which are solved using the Fundamental Theorem of Calculus.
• In this situation, because the upper limit is infinity, we set up an expression using the limit method to handle the improper nature of the integral.

Definite integration requires evaluating the antiderivative at these boundaries, subtracting the result from one boundary from the other. This helps in getting the exact result of the total accumulation across that particular interval. Understanding this will aid immensely in solving problems involving areas, volumes, and other quantities which can be derived from integrals.

Antiderivative

The antiderivative, sometimes called the indefinite integral, is essential for finding integrals. It reverses the action of finding a derivative. In the exercise solution, the function \(3x^{-2}\) required determining its antiderivative to solve the integral.

• To obtain the antiderivative, we look for a function whose derivative gives us the original function \(3x^{-2}\).
• In this case, the antiderivative of \(3x^{-2}\) is \(-\frac{3}{x}\), because differentiating \(-\frac{3}{x}\) gives back \(3x^{-2}\).

Once we have the antiderivative, we apply it over the limits of the integral to find the definite integral's value. This process is critical as it forms the basis of integrating functions and solving complex calculus problems.

Limit Evaluation

Limit evaluation is a powerful tool used when dealing with improper integrals, allowing us to assess the behavior of functions at infinity or other undefined points. In our example, we had to evaluate the integral from 10 to infinity, which inherently involves dealing with a limit.

• To evaluate an improper integral like this one, we replace the infinity symbol with a variable (usually \(b\)) and then evaluate the integral over a new range from 10 to \(b\).
• After calculating the definite integral, we take the limit as \(b\) approaches infinity.

In this particular case, as \(b\) approaches infinity, the term \(-\frac{3}{b}\) becomes zero, simplifying the expression to \(\frac{3}{10}\). This demonstrates the importance of limit evaluation in finding a meaningful, finite result for an otherwise infinite calculation, ensuring the integrity and applicability of solutions derived from improper integrals.