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 2-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out. $$ \begin{array}{|l|c|c|c|c|c|c|} \hline t(\mathrm{~h}) & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline r(t)(\mathrm{L} / \mathrm{h}) & 8.7 & 7.6 & 6.8 & 6.2 & 5.7 & 5.3 \\ \hline \end{array} $$

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 2-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline t(\mathrm{~h}) & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline r(t)(\mathrm{L} / \mathrm{h}) & 8.7 & 7.6 & 6.8 & 6.2 & 5.7 & 5.3 \\
\hline
\end{array}
$$

Key Concepts

Monotonic Decreasing Functions

A monotonic decreasing function is one that consistently decreases as its input increases. This property is important in the context of Riemann sums because it determines the relationship between the left endpoint and right endpoint evaluations. Specifically, for such functions, the left endpoint evaluation in each subinterval will be higher than the right endpoint evaluation, aiding in the estimation of bounds for the integrated value.

Left and Right Endpoint Approximations

When applying Riemann sums, the choice of endpoints affects whether the approximation overestimates or underestimates the true value. For a decreasing function, using the left endpoints of subintervals typically yields an overestimate (upper sum), while using the right endpoints gives an underestimate (lower sum). This concept is fundamental in obtaining error bounds for the approximation.

Definite Integral

A definite integral represents the limit of a sum of infinitesimal quantities and is used to calculate the accumulation of a quantity, such as the total amount of oil leaked when the leakage rate is known as a function of time. It essentially sums up the 'area under the curve' of the rate function over a period.

Riemann Sums

Riemann sums are methods for approximating the value of a definite integral by breaking the integration interval into smaller subintervals. In this approach, the function values at chosen sample points (such as the left or right endpoints) within each subinterval are multiplied by the width of the subinterval to approximate the area, which is particularly useful when the exact integral is difficult to compute.

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