Let r be the region bounded by the x-axis, the graph of y = x + 1, and the line x = 3. Find the area of the region r.

To find the area of the region r, we need to first understand the boundaries defined in the question. The region is bounded by the x-axis, the line x = 3, and the line represented by the equation y = x + 1.

The first step is to identify the points where the line y = x + 1 intersects the x-axis. This occurs when y = 0. Setting the equation y = x + 1 to 0 gives us:

0 = x + 1

From this, we can solve for x:

x = -1

Now, we know that the region we’re interested in is between x = -1 and x = 3, bordered below by the x-axis and above by the line y = x + 1. To find the area, we can set up the integral of the function y = x + 1 from x = -1 to x = 3:

Area = ∫ from -1 to 3 (x + 1) dx

Now, we can calculate this integral:

Area = [1/2 * x^2 + x] from -1 to 3

Calculating at the bounds:

Area = [(1/2 * 3^2 + 3) - (1/2 * (-1)^2 + (-1))]

Simplifying this gives:

Area = [(1/2 * 9 + 3) - (1/2 * 1 - 1)]

Area = [(4.5 + 3) - (0.5 - 1)]

Area = [7.5 + 0.5] = 8

Therefore, the area of the region r is 8 square units.