Let {{MATH_47}} be the velocity of some object at time {{MATH_48}}. {{CMD_49}} {{ENV_50}} {{CMD_51}} and let {{MATH_52}} be an antiderivative of {{MATH_53}}. Suppose we want to approximate {{MATH_54}}. Let's get an approximation using {{MATH_55}} subintervals. {{ENV_56}} We can visualize this as the sum of the areas of {{MATH_57}} rectangles. Note that while the first {{MATH_58}} terms are positive, the last is negative because {{MATH_59}}. We should think of these rectangles as having {{CMD_60}}signed area{{CMD_61}}, where the area is negative when it lives below the {{MATH_62}}-axis.