Key Concepts and Formulas

• Area between a curve and the x-axis: If $f(x) \ge 0$ on $[a, b]$, the area $A = \int_a^b f(x) \, dx$.
• Area between two curves: If $f(x) \ge g(x)$ on $[a, b]$, the area $A = \int_a^b (f(x) - g(x)) \, dx$.
• Splitting Regions: Divide the region into sub-regions where the upper and lower bounding curves are consistent.

Step-by-Step Solution

Step 1: Understand the Given Curves and the Region

We are given the curves $y = x$, $x = e$, $y = \frac{1}{x}$, and the positive x-axis ($y=0$). We need to find the area of the region enclosed by these curves.

• $y = x$ is a straight line.
• $x = e$ is a vertical line at $x \approx 2.718$.
• $y = \frac{1}{x}$ is a hyperbola.
• The positive x-axis acts as a lower bound.

Step 2: Determine Intersection Points

Finding the intersection points helps define the limits of integration.

• Intersection of $y = x$ and $y = \frac{1}{x}$: $x = \frac{1}{x} \implies x^2 = 1$. Since we are in the positive x-axis, $x = 1$. Thus, $y = 1$, and the intersection point is $(1, 1)$.

• Intersection of $y = \frac{1}{x}$ and $x = e$: $y = \frac{1}{e}$. The intersection point is $(e, \frac{1}{e})$.

• Intersection of $y = x$ and $x = e$: $y = e$. The intersection point is $(e, e)$.

Step 3: Sketch the Region and Divide into Sub-regions

A sketch is crucial. The region is bounded by the x-axis, $y=x$ from $x=0$ to $x=1$, and $y=\frac{1}{x}$ from $x=1$ to $x=e$. Therefore, we need to split the region into two parts:

• Region 1: Bounded by $y = x$, $y = 0$, $x = 0$, and $x = 1$.
• Region 2: Bounded by $y = \frac{1}{x}$, $y = 0$, $x = 1$, and $x = e$.

Step 4: Calculate the Area of Region 1 ($A_1$)

The area of Region 1 is given by: $A_1 = \int_0^1 x \, dx$ $A_1 = \left[ \frac{x^2}{2} \right]_0^1$ $A_1 = \frac{1^2}{2} - \frac{0^2}{2} = \frac{1}{2}$

Step 5: Calculate the Area of Region 2 ($A_2$)

The area of Region 2 is given by: $A_2 = \int_1^e \frac{1}{x} \, dx$ $A_2 = \left[ \ln|x| \right]_1^e$ $A_2 = \ln(e) - \ln(1) = 1 - 0 = 1$

Step 6: Calculate the Total Area

The total area $A$ is the sum of $A_1$ and $A_2$: $A = A_1 + A_2 = \frac{1}{2} + 1 = \frac{3}{2}$

Common Mistakes & Tips

• Sketching: Always sketch the region to correctly identify the limits of integration and the upper/lower functions.
• Intersection Points: Calculate intersection points accurately.
• Splitting Regions: Correctly identify where the bounding curves change and split the integral accordingly.

Summary

We found the area of the region enclosed by the given curves by splitting it into two sub-regions and calculating the area of each using definite integrals. The total area is the sum of the areas of the sub-regions, which is $\frac{3}{2}$ square units.

The final answer is \boxed{3/2}, which corresponds to option (B).