Problem 23 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\ln x, y=0\), and \(x=e\).

Step-by-Step Solution

Answer

The region R is described by the inequalities \(1 \leq x \leq e\) and \(0 \leq y \leq \ln x\).

Step 1: Identify the boundaries

The region R is bounded by the curves and lines given in the problem. Specifically: 1. The curve given by the natural logarithm function, \(y = \ln x\).2. The horizontal line \(y = 0\) (the x-axis).3. The vertical line \(x = e\).

Step 2: Express the vertical cross sections

To describe R in terms of vertical cross sections, observe that for any x in the interval \( (1, e] \), the corresponding y-values are bounded between the curve \(y = \ln x\) and the x-axis. Therefore, the inequalities for y given a fixed x are: \[ 0 \leq y \leq \ln x \]

Step 3: Express the horizontal cross sections

To describe R in terms of horizontal cross sections, observe that for any y in the interval \( [0, 1] \), the corresponding x-values are bounded between the vertical line at x = 1 and the exponential curve. Therefore, solving \(y = \ln x\) for x gives:\( x = \exp(y) \). Thus, the inequalities for x given a fixed y are: \[ 1 \leq x \leq e^y \]

Step 4: Combine the inequalities

The final step is to combine the vertical and horizontal cross section descriptions into a full characterization of the region R: The entire region can be described as: \[ 1 \leq x \leq e \] and \[ 0 \leq y \leq \ln x \]

Key Concepts

bounded region

When we talk about a 'bounded region' in calculus, we are describing an area on a graph that is contained within certain boundaries. These boundaries can be lines or curves. For this exercise, the region R is bounded by three items:
• The natural logarithm curve: \( y = \ln x \)
• The horizontal line: \( y = 0 \) (also known as the x-axis)
• The vertical line: \( x = e \)
Imagine drawing these on a graph. The curve \( y = \ln x \) rises to the right, starting from the point \( x = 1 \) and going through the point \( x = e \). The line \( y = 0 \) runs horizontally at the base. The line \( x = e \) stands upright and intersects the curve and the horizontal line. Together, they form a closed shape that encapsulates the region R.

natural logarithm

The natural logarithm, denoted as \( \ln x \), is a special function in mathematics. It is the inverse of the exponential function \( e^x \). This means that if you have \( y = \ln x \), you can also write \( x = e^y \). The natural logarithm function has some interesting properties:
• It is only defined for positive values of x \((x > 0)\).
• It passes through the point \( (1, 0) \) because \( \ln 1 = 0 \).
• As x increases, the value of \( \ln x \) also increases, but at a slower rate.
In the context of the exercise, the curve \( y = \ln x \) is one of the boundaries of the region R. For any given x-value within the interval \(1 \leq x \leq e\), the y-value starts at 0 and goes up to \( \ln x \).

cross sections

Cross sections help to describe a region by cutting it into thinner slices. There are two types of cross sections to consider: vertical and horizontal.

Vertical Cross Sections:
When we take vertical cross sections, we fix a certain x-value and look at the possible y-values within the region for that x. For the region R in this exercise, a fixed x means our cross section runs between \( y = 0 \) and \( y = \ln x \). For any x in the interval \(1 \leq x \leq e\), the inequality is:
\[ 0 \leq y \leq \ln x \]

Horizontal Cross Sections:
For horizontal cross sections, we fix a y-value and look at the x-values contained within the region. In our example, solving \( y = \ln x \) for x gives us \( x = e^y \). Thus, for any y in the interval \(0 \leq y \leq \ln e = 1\), we get:
\[ 1 \leq x \leq e^y \]