Counting Lattice Points
To find the number of lattice points on the boundary and inside the region bounded by the x-axis, x = 4, and y = x^2, we need to carefully analyze the given region and count the lattice points within it.

Region Analysis
The region is bounded by the x-axis, x = 4, and y = x^2. Let's break down the region into smaller parts for analysis:

1. Section A: The region between the x-axis and x = 1
- In this section, the values of x range from 0 to 1, inclusive.
- The y-values are given by y = x^2, so the possible y-values in this section are 0 and 1.
- Therefore, there are 2 lattice points in this section: (0, 0) and (1, 1).

2. Section B: The region between x = 1 and x = 4
- In this section, the values of x range from 1 to 4, inclusive.
- The y-values are given by y = x^2, so the possible y-values in this section are 1, 4, 9, and 16.
- Among these y-values, we need to identify the lattice points.
- The lattice points in this section are: (1, 1), (2, 4), (3, 9), and (4, 16).

3. Section C: The region between the x-axis and x = 4
- In this section, the values of x range from 0 to 4, inclusive.
- The y-values are given by y = x^2, so the possible y-values in this section are 0, 1, 4, 9, and 16.
- Among these y-values, we need to identify the lattice points.
- The lattice points in this section are: (0, 0), (1, 1), (2, 4), (3, 9), and (4, 16).

Counting Lattice Points
Now that we have identified the lattice points in each section, we can count the total number of lattice points:

- Section A: 2 lattice points
- Section B: 4 lattice points
- Section C: 5 lattice points

Therefore, the total number of lattice points on the boundary and inside the region is 2 + 4 + 5 = 11.

However, we need to include the lattice points on the x-axis as well. In this case, the lattice points on the x-axis are (0, 0), (1, 0), (2, 0), (3, 0), and (4, 0). These points are not included in the previous count.

Adding the lattice points on the x-axis to the previous count, we get 11 + 5 = 16.

Hence, the correct answer is 16.

Correction of the Answer
Apologies for the previous incorrect answer. After revisiting the analysis, we identified an error in counting the lattice points. Let's correct it:

- Section A: 2 lattice points
- Section B: 4 lattice points
- Section C: 5 lattice