The rate of a certain country’s exports is given by: ??(??) = ??. ?????? ?? - ??. ???? + ??. ?? billion dollars per year,

Question

The rate of a certain country’s exports is given by: ??(??) = ??. ?????? ?? − ??. ???? + ??. ?? billion dollars per year, and its rate of imports is given by ??(??) = −??. ?????? + ??. ?? billion dollars per year, where t is the number years since 1990. What is the difference in total accumulation of the values of the country’s imports and exports from 1995 to 2000? In other words, does the country import more or export more over that time period, and by how much?

Answer 1

The rate of exports is the derivative of the amount of exports, that is if E(t) is the amount of exports at time t then E'(t) is the rate of exports at time t.  So the accumulation of exports is ∫₅¹⁰ E'(t)dt which is equal to E(10) - E(5) by the Fundamental Theorem of Calculus since E is an anti-derivative of E'.  

 

Similarly for imports the net change is ∫₅¹⁰ I'(t)dt = I(10) - I(5)