Calculus

A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water using the Midpoint Rule with m = 2 and n = 3.

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  1. Divide the surface of the pool into 4x6 = 24 rectangular columns.
    Consider the depth of each column to be the value at the center of the x- and y-intervals.
    So, picking a column where all four corners are at different depths, to illustrate finding the depth at the center of the rectangular column,
    Pick the x-interval (10,15) and the y-interval (10,15). Now, we do not know what the depth is at (12.5,12.5), and the four corners on the bottom do not necessarily all lie in the same plane, but we can use the average of the depths at the centers of the diagonals, which would be
    ((6+5)/2 + (8+4)/2)/2 = (13/2 + 12/2)/2 = 25/4
    So, calculate the depth at the center of each rectangle and add them up, and then multiply the result by 25, the area of each rectangle.

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    oobleck

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