The Riemann sum is based on a simple concept: approximating the area under a graph using rectangles. The area of a rectangle is length times width, so width is the difference between two close values of x while length, or height, is either the value of the function for the lower value of x or the higher. The rectangle that sits on the right above the graph will be an overestimate of the area while the rectangle sitting on the left under the graph will be an underestimate.
There are 6 points so there will be 5 rectangles. If we take the first rectangle we have a right hand value of x of 3 and a right-hand value of f(x) of -2. So the width of the rectangle along the x axis is 3 and the height below the x axis is 2 making the area 2*3=6. Rectangle 1 has area 6.
Rectangle 2 has width 5-3=2 and height=1 making the area=2.
Rectangle 3 has width 9-5=4 and height=0 making area=0.
Rectangle 4 has width 13-9=4 and height=1 making area=4.
Rectangle 5 has width 14-13=1 and height=0 making area=0.
Add these areas together: 6+2+0+4+0=12 square units.