**Problem Analysis:**
We are given that a circle is tangent to the x-axis at 2 and the y-axis at 2. We need to find the area of the region between the circle and the origin.

**Solution:**
Let's assume the center of the circle is (a, b) and the radius is r.

**Step 1: Finding the equation of the circle**
Since the circle is tangent to the x-axis at 2, we know that the distance between the center of the circle and the x-axis is equal to the radius of the circle. So, |b - 0| = r, which means b = r.

Similarly, since the circle is tangent to the y-axis at 2, we know that the distance between the center of the circle and the y-axis is equal to the radius of the circle. So, |a - 0| = r, which means a = r.

Therefore, the equation of the circle can be written as (x - a)^2 + (y - b)^2 = r^2, which simplifies to (x - r)^2 + (y - r)^2 = r^2.

**Step 2: Finding the intersection points**
To find the intersection points of the circle with the x-axis and the y-axis, we can substitute y = 0 and x = 0 in the equation of the circle.

When y = 0, we get (x - r)^2 + (0 - r)^2 = r^2, which simplifies to (x - r)^2 + r^2 = r^2. This equation simplifies to (x - r)^2 = 0, which means x - r = 0. So, x = r.

When x = 0, we get (0 - r)^2 + (y - r)^2 = r^2, which simplifies to r^2 + (y - r)^2 = r^2. This equation simplifies to (y - r)^2 = 0, which means y - r = 0. So, y = r.

Therefore, the intersection points are (r, 0) and (0, r).

**Step 3: Finding the area between the circle and the origin**
To find the area between the circle and the origin, we need to find the area of the sector of the circle formed by the points (r, 0) and (0, r), and subtract the area of the triangle formed by the points (0, 0), (0, r), and (r, 0).

The area of the sector can be found using the formula A = (1/2)r^2θ, where θ is the central angle. Since the sector is formed by the points (r, 0) and (0, r), the central angle θ is 90 degrees or π/2 radians.

So, the area of the sector is A = (1/2)r^2(π/2) = (π/4)r^2.

The area of the triangle can be found using the formula A = (1/2)bh, where b is the base and h is the height. In this case, the base is r and the height is r. So, the area of the triangle is A = (1/2)r*r = (1/2