Oh mein lieber Gott, make me unsee it! Integration was invented only in 1994, in case you didn't know:
Tai, "A mathematical model for the determination of total area under glucose tolerance and other metabolic curves.", Diabetes Care 17, 152 (1994).
Some phrases from the abstract:
...In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve...
....The Tai model allows flexibility in experimental conditions, which means, in the case of the glucose-response curve, samples can be taken with differing time intervals and total area under the curve can still be determined with precision.