On the one hand, we can interpret each {{MATH_65}} as an approximation in the change in the height of {{MATH_66}} over the subinterval. On the other hand, since {{MATH_67}}, we can think of {{MATH_68}} as the area of a rectangle based on the subinterval, and with hieght {{MATH_69}}. So the total change in {{MATH_70}} over {{MATH_71}} can be reinterpreted as the sum of the signed areas of the rectangles under {{MATH_72}}. If we make our subintervals smaller, the repeated linear approximation should get more precise, and the sum of the areas of the rectangles gets closer to the area under the curve: