Oil leaked from a tank at a rate of r(t) liters per hour....

Oil leaked from a tank at a rate of $r(t)$ liters per hour. The rate decreased as time passed and values of the rate at twohour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
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Key Concepts

Monotonic Functions and Endpoint Approximations

For functions that are monotonic (either always increasing or always decreasing), specific endpoint choices yield systematic under- or overestimates. When a function is decreasing, as in this context, using the left endpoint of each subinterval provides an overestimate (upper sum), while using the right endpoint gives an underestimate (lower sum). This property allows one to construct bounds for the total accumulated quantity.

Riemann Sum Approximations

Riemann sums are a method for approximating the definite integral of a function by summing the product of function values and the widths of subintervals. The approximation becomes more accurate as the subintervals become smaller. In practical applications, especially when using tabulated data, the integral is estimated by summing values at specific sample points multiplied by the interval width.

Rate Function and Integration

A rate function describes how a quantity changes over time. When evaluating the total accumulated change, such as the total amount of oil leaked, integration is used to sum up the product of the rate and the infinitesimal time intervals over the period of interest. This integral represents the total quantity accumulated from a continuously varying rate.

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