Find the area of the region in the first quadrant bounded by the line yequals3​x, the line xequals4​, the curve yequalsStartFraction 3 Over x EndFraction
​, and the​ x-axis.
Question content area bottom
Part 1
The total area of the region is 
  
enter your response here.
​(Type an exact​ answer, using radicals as​ needed.)

To find the area of the region in the first quadrant bounded by the given lines and curves, let's break the problem down into steps.

### 1. Identify the curves and boundaries:
- Line \( y = 3x \)
- Line \( x = 4 \)
- Curve \( y = \frac{3}{x} \)
- \( x \)-axis (i.e., \( y = 0 \))

### 2. Find the points of intersection:
We need to find the points where these curves intersect each other to determine the boundaries of the region.

- Intersection of \( y = 3x \) and \( y = \frac{3}{x} \):
  
  Set \( 3x = \frac{3}{x} \):
  \[
  3x^2 = 3 \implies x^2 = 1 \implies x = 1
  \]
  For \( x = 1 \), \( y = 3 \cdot 1 = 3 \). Thus, the point of intersection is \( (1, 3) \).

- Intersection of \( y = 3x \) and \( x = 4 \):

  When \( x = 4 \), \( y = 3 \cdot 4 = 12 \). So, the point of intersection is \( (4, 12) \).

- Intersection of \( y = \frac{3}{x} \) and \( x = 4 \):
  
  When \( x = 4 \), \( y = \frac{3}{4} \). So, the point of intersection is \( (4, \frac{3}{4}) \).

### 3. Set up the integrals for the area:
The area can be found by integrating the difference between the curves \( y = 3x \) and \( y = \frac{3}{x} \) from \( x = 1 \) to \( x = 4 \).

The area \( A \) is given by:
\[
A = \int_{1}^{4} \left(3x - \frac{3}{x}\right) \, dx
\]

### 4. Compute the integral:
\[
\int_{1}^{4} 3x \, dx = \left[ \frac{3x^2}{2} \right]_{1}^{4} = \frac{3(4^2)}{2} - \frac{3(1^2)}{2} = \frac{3(16)}{2} - \frac{3}{2} = 24 - \frac{3}{2} = \frac{48 - 3}{2} = \frac{45}{2}
\]

\[
\int_{1}^{4} \frac{3}{x} \, dx = 3 \ln |x| \bigg|_{1}^{4} = 3 \ln 4 - 3 \ln 1 = 3 \ln 4
\]

### 5. Final area calculation:
\[
A = \frac{45}{2} - 3 \ln 4
\]

So, the exact area of the region is:
\[
\boxed{\frac{45}{2} - 3 \ln 4}
\]

Would you like further details on any steps or have any questions?

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Here are some related questions:
1. How do you determine the points of intersection between two curves?
2. What is the geometric interpretation of the integral in this context?
3. How do you evaluate definite integrals involving logarithms?
4. How would the solution change if one of the curves was different?
5. What methods can be used to verify the calculated area?

**Tip:** When dealing with regions bounded by multiple curves, sketching the graph can help in visualizing the problem and setting up the correct integrals.

Learn how to calculate the area of a region bounded by lines and curves in the first quadrant using calculus. This problem involves integrating between intersecting curves and lines, including the use of logarithms for evaluation. Suitable for advanced high school students.

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