An integral is the limit of weighted samples as the partition becomes fine. On , the decreasing function makes left-endpoint rectangles overestimate the area; the excess tends to zero.积分是分割不断加细时加权采样和的极限。在 上， 单调递减，因此左端点矩形高估曲线下的面积；多出的面积趋于零。
The green rectangles use 4, 8, 16, and 32 intervals. Their top-left corners meet the gold curve, whose exact enclosed area is .绿色矩形依次使用 4、8、16 和 32 个区间，左上角落在金色曲线上。曲线下的准确面积为 。