find the area of the region bounded by y=2x, the x-axis, and x=4

To find the area of the region bounded by the line \( y = 2x \), the x-axis, and the vertical line \( x = 4 \), we need to calculate the area under the curve of \( y = 2x \) from \( x = 0 \) to \( x = 4 \).

This is done by integrating the function \( y = 2x \) with respect to \( x \), over the interval \( [0, 4] \):

\[
A = \int_{0}^{4} 2x \, dx
\]

Now, we compute the integral:

\[
A = \left[ x^2 \right]_{0}^{4} = 4^2 - 0^2 = 16
\]

Thus, the area of the region is 16 square units.

Do you want further details on how the integration works? Here are 5 related questions you can explore:

1. How would the area change if the line were \( y = 3x \) instead of \( y = 2x \)?
2. What is the geometric interpretation of this problem?
3. How do you calculate areas under curves for non-linear functions?
4. How does the integral change if the limits of integration are different?
5. What happens if the line crosses the x-axis in this type of problem?

**Tip:** When integrating to find areas, make sure the function stays above the x-axis or break it into sections for proper evaluation.

Find the area of the region bounded by y=2x, the x-axis, and x=4.

This solution explains how to find the area of the region bounded by the line y=2x, the x-axis, and the vertical line x=4. The problem is solved using definite integration, covering key calculus concepts such as area under a curve and the Fundamental Theorem of Calculus. Suitable for advanced high school students.

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