Question

Oil leaks out of a tanker at a rate of r=f(t) liters per minute, where t is in minutes. If f(t) = A e^{-k t}, write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour. total quantity = \int_{a}^{b} (where a = and b = ). What are the units of this integra

Answer 1

Answer:

$V_{total} = \displaystyle\int_0^{60} A e^{-k t}~dt$

Step-by-step explanation:

We are given the following in the question:

Oil leaks out of a tanker at a rate of r = f(t) liters per minute, where t is in minutes.

$f(t) = A e^{-k t}$

Let V be the volume, then we are given that rate of leakage is:

$\displaystyle\frac{dV}{dt} = f(t) = A e^{-k t}$

Thus, we can write:

$dV = f(t).dt = A e^{-k t}~dt$

We have to find the  a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

Thus, total amount of oil leaked will be the definite integral from 0 minutes to 60 minutes.

$dV =A e^{-k t}~dt\\V_{total} = \displaystyle\int_a^b f(t)~dt\\\\a = 0\text{ minutes}\\b = 60\text{ minutes}\\\\V_{total} = \displaystyle\int_0^{60} A e^{-k t}~dt$

The units of integral will be liters.