Integration by parts is one of the best because it is used when a function that has to be integrated is written as a product of two or more.
The oil leaks out of the tanker from the time t = 0 to t = 10 is 4000.79 gallons.
Given
Oil is leaking from a tanker at the rate of [tex]\rm R(t) = 2000e^{(-0.2t)}[/tex] gallons per hour where t is measured in hours.
What is integration?
Integration by parts is one of the best because it is used when a function that has to be integrated is written as a product of two or more.
The total amount of oil that leaks out over some amount of time can be found by integrating f(t) over the interval of time.
This of the integral is the sum of each of the amounts when we cut the time up in seconds, or milliseconds though in reality, the intervals are so small they are practically zero, this is why we use an integral rather than a simple sum.
Integrating the function from t =0 to t = 10.
[tex]\rm R(t) =\int\limits^{10}_0 { 2000e^{(-0.2t)}} dt\\\\R(t) = 2000\int\limits^{10}_0e^{-0.2t} dt\\\\R(t) = 2000 [(-1) \times e^{-0.2t}]^{10}_0\\\\R(t) = -2000[e^{-0.2t}]^[10}_0\\\\R(t) = -2000[e^{0.2(10)]}\\\\R(t) = -2000 \times {e^2}\\\\R(t) = -2000 \times 2.79\\\\R(t) = 4000.79[/tex]
Hence, the oil leaks out of the tanker from the time t = 0 to t = 10 is 4000.79 gallons.
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