Suppose that water is flowing into a tank at a rate of r (t) gallons per hour, where the rate depends on the time t according to the formula

(a) Consider a brief period of time, say, from t₁ to t₂. The length of this time period is t = t₂ - t₁. During this period the rate of flow does not change much and is approximately 20 - 4t₁ (the rate at the beginning of the brief time interval). Approximately how much water flows into the tank during the time from t₁ to t₂ ?

(b) Explain why the total amount of water added to the tank during the time interval from t = 0 to t = 5 is given by ∫₀⁵ r(t)dt.

r(t) = 20 - 4t, 01 5.