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The problem asks for the area of the region bounded by the curve \( y = 2x \), the x-axis, and the vertical lines \( x = 2 \) and \( x = 4 \).

### Solution:
To find the area of the region bounded by these curves, you can use the definite integral. The area \( A \) under the curve \( y = 2x \) from \( x = 2 \) to \( x = 4 \) is given by:

\[
A = \int_{2}^{4} 2x \, dx
\]

1. **Set up the integral**:
   \[
   A = \int_{2}^{4} 2x \, dx
   \]

2. **Integrate**:
   The integral of \( 2x \) with respect to \( x \) is:
   \[
   \int 2x \, dx = x^2
   \]

3. **Evaluate the definite integral**:
   \[
   A = \left[ x^2 \right]_{2}^{4}
   \]
   This means you substitute the upper limit (4) and the lower limit (2) into the integrated function:
   \[
   A = (4^2) - (2^2)
   \]
   \[
   A = 16 - 4 = 12
   \]

So, the area of the region is **12 square units**.

Would you like to see the details or ask any questions about this solution?

Here are five related questions to further your understanding:
1. How would you calculate the area if the line was \( y = 3x \) instead of \( y = 2x \)?
2. What if the region was bounded by \( y = 2x \) and the line \( x = 6 \)? How would the area change?
3. Can you find the area of the region between the curve \( y = 2x \) and the line \( y = 8 \) over the same interval?
4. How would the calculation change if the x-axis was shifted to \( y = 1 \)?
5. What is the geometric interpretation of this area calculation?

**Tip**: When integrating a function to find the area under a curve, the limits of integration correspond to the x-values where the region begins and ends.

Solve number 68: Find the area of the region bounded by y = 2x, the x-axis, and the vertical lines x = 2 and x = 4.

Learn how to calculate the area of the region bounded by y = 2x, the x-axis, and vertical lines x = 2 and x = 4 using definite integrals. This step-by-step guide covers key calculus concepts, including the Fundamental Theorem of Calculus, and is suitable for students in Grades 11-12.

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