If we use the left edge of the time slice as the height of each rectangle the area we get by summing over the rectangles will be under the actual value by the sum of the area of the small triangles under the velocity curve. If we use the right edge of the time slice as the height, the area will be over the actual value by the sum of the area of the small triangles above the velocity curve. Since the velocity "curve" is in this case a straight line we could solve this problem by using the mid-value but it will be useful for us to have a way to get the area under a general curve so we will take a less direct approach.
Suppose we use the right edge estimate of the area under the curve. How much in error is it? Let's take all four of the little triangles which contribute to the error and pile them all together. Click on the Action button to see this.