Area bounded by the curve ( y=ln x, y=0 ) and ( x=3 ) is

To find the area bounded by the curve \( y = \ln x \), the x-axis \( y = 0 \), and the vertical line \( x = 3 \), we need to evaluate the definite integral of the function \( y = \ln x \) from \( x = 1 \) to \( x = 3 \).

The area \( A \) can be expressed as:

\[
A = \int_{1}^{3} \ln(x) \, dx
\]

To solve this, we'll integrate \( \ln(x) \):

1. Use integration by parts:
   \[
   \int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx
   \]
   \[
   \int \ln(x) \, dx = x \ln(x) - \int 1 \, dx
   \]
   \[
   \int \ln(x) \, dx = x \ln(x) - x + C
   \]

2. Now, apply the limits \( x = 1 \) to \( x = 3 \):
   \[
   A = \left[ x \ln(x) - x \right]_{1}^{3}
   \]
   \[
   A = \left[ 3 \ln(3) - 3 \right] - \left[ 1 \ln(1) - 1 \right]
   \]

3. Simplify:
   \[
   A = \left( 3 \ln(3) - 3 \right) - (0 - 1)
   \]
   \[
   A = 3 \ln(3) - 2
   \]

Thus, the area bounded by the curve \( y = \ln x \), the x-axis, and \( x = 3 \) is \( 3 \ln(3) - 2 \).

Would you like a detailed explanation of integration by parts or have any other questions?

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Here are 8 related questions to expand your understanding:
1. What is the area bounded by \( y = \ln x \), \( y = 0 \), and \( x = 2 \)?
2. How does the area change if the curve is \( y = e^x \) instead of \( y = \ln x \)?
3. Can you explain the geometric interpretation of integration by parts?
4. What is the integral of \( y = \ln(x^2) \)?
5. How would you find the volume of a solid formed by rotating \( y = \ln x \) around the x-axis?
6. How does changing the limits of integration affect the area calculation?
7. What is the significance of the natural logarithm in calculus?
8. How do you calculate the area between two curves?

**Tip:** When solving area problems bounded by curves, always check if the curve intersects the axis or other curves within the given limits to ensure correct setup of the integral.

Learn how to calculate the area bounded by the curve y = ln(x), the x-axis, and the vertical line x = 3 using definite integrals and integration by parts. Detailed step-by-step solution provided. Suitable for advanced high school students and calculus enthusiasts.

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