Problem 26 - Calculus and its applications

Question

Find the area, if it is finite, of the region under the graph of \(y=1 / x\) over \([2, \infty)\).

Step-by-Step Solution

Answer

The area is infinite.

Step 1: Understanding the Problem

We need to find the area under the curve of the function \( y = \frac{1}{x} \) from \( x = 2 \) to \( x = \infty \). This involves calculating the integral of the function over this interval.

Step 2: Setting Up the Integral

The area under the curve from \( x = 2 \) to \( x = \infty \) is given by the improper integral \( \int_{2}^{\infty} \frac{1}{x} \, dx \).

Step 3: Evaluating the Integral

Since the integral is improper due to the upper limit being infinity, we use limits to evaluate it: \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{x} \, dx \].

Step 4: Calculating the Definite Integral

The antiderivative of \( \frac{1}{x} \) is \( \ln |x| \). Thus, we have: \( \int_{2}^{b} \frac{1}{x} \, dx = \left[ \ln |x| \right]_{2}^{b} = \ln |b| - \ln 2 \).

Step 5: Taking the Limit

We now consider the limit as \( b \) approaches infinity: \[ \lim_{b \to \infty} (\ln |b| - \ln 2) = \lim_{b \to \infty} \ln |b| - \ln 2 = \infty - \ln 2 = \infty \].

Step 6: Conclusion

The area is infinite, meaning no finite area exists under the curve \( y = \frac{1}{x} \) from \( x = 2 \) to \( x = \infty \).

Key Concepts

Area under a curve

The concept of the "area under a curve" is fundamental in calculus. It allows us to find the space between a curve and the x-axis over a specified interval. This area can represent physical quantities, like distance or mass, depending on the context. To find the area, we often use integrals. An integral sums infinite small parts of the area, which, when added together, give the total area under the curve.
The function in our example is a classic case where the area under the curve from a starting point to infinity needs to be calculated. This is done using an improper integral, making sure we calculate limits correctly. The determination of this area leads us to the understanding of potentially finite or infinite values based on the behavior of the function and its limits.

Limit of a function

Limits help us understand the behavior of functions as they approach a given point or infinity. When dealing with improper integrals, limits are crucial.
In the context of the given exercise, as we calculate the area under the curve of \( y = \frac{1}{x} \) from 2 to infinity, we set up a limit to manage the infinity part. This limit evaluates the behavior of the integral as its upper bound extends to infinity. Here, \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{x} \, dx \] is used.
The evaluation of limits is essential to determine whether an integral like this converges (results in a finite area) or diverges (results in an infinite area). In our exercise, the limit results in infinity, confirming the area is infinite.

Definite Integrals

Definite integrals compute the accumulated quantity, such as area, between two bounds. They are represented as \[ \int_{a}^{b} f(x) \, dx, \] where \(a\) and \(b\) are the lower and upper bounds, respectively. The integration process gives us the total accumulation of the function \(f(x)\) over the interval \([a, b]\).
In the scenario of an improper integral, like our example with \( y = \frac{1}{x} \), the upper bound of the interval is replaced by a limit. This turns the definite integral into \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{x} \, dx. \]
This integral, after finding its antiderivative, allows us to evaluate the function's total behavior from a finite starting point towards an infinite distance.

Antiderivatives

Antiderivatives, also known as indefinite integrals, help us find the original function whose derivative matches a given function. The process is essentially the reverse of differentiation and is key to evaluating integrals. For our function \( y = \frac{1}{x} \), the antiderivative is \( \ln |x| \).
Once we have the antiderivative, we use it to compute definite integrals by evaluating it at the bounds. In our case, \[ \int_{2}^{b} \frac{1}{x} \, dx = \ln |b| - \ln 2. \]
This expression shows the net effect of the function from one point to another. Using limits with antiderivatives allows us to handle functions approaching infinity, as it did here when determining if the area was finite or infinite under our curve towards infinity.