find the area A of the region bounded by the gragh of y=x+1,the x-axis,and the lines x=2 and x=4

We have:

4                                  

∫ x + 1  dx   =    (1/2)[16 - 4] + [4 - 2]  =  6 + 2  = 8 square units

2

Here's the graph.......https://www.desmos.com/calculator/erdardlwlm

Notice that we could also do this without Calculus......the small triangle formed by the bounded region  y=4, x = 4 and y = x + 1   equals the same triangular region bounded by y = 4, x = 2 and y= x + 1. And adding this second region to the region bounded by y= x + 1, y = 0, y = 4, x = 2 and x = 4, we have a rectangle that is 2 units wide and 4 units high.......thus giving us an area of 8 square units.....!!!!