Question

explain in words what the integral represents and give units.
\\( \int _ { 1986 } ^ { 2008 } f ( t ) d t \\),
where \\( f ( t ) \\) is the rate at which the worlds population is growing in year \\( t \\), in billions of people per year.
the integral represents the
the units are
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Explanation:

The integral of a rate function \( f(t) \) over an interval \([a,b]\) (here \(a = 1986\) and \(b=2008\)) gives the net - change of the quantity that the rate function describes. Since \(f(t)\) is the rate of population growth (in billions of people per year), integrating it over the time interval from 1986 to 2008 will give the total change in the world's population.

For the units, when we integrate a function \(y = f(x)\) (in this case \(y=f(t)\) with \(y\) having units of billions of people per year and \(x=t\) having units of years), using the formula \(\int_{a}^{b}f(x)dx\approx\sum_{i = 1}^{n}f(x_{i})\Delta x\) (Riemann sum). The units of \(f(x_{i})\) is (billions of people/year) and the units of \(\Delta x\) (or \(dt\)) is year. So, (billions of people/year)\(\times\)year = billions of people.

Answer:

The integral \(\int_{1986}^{2008}f(t)dt\) represents the total change in the world's population between the years 1986 and 2008. The units are billions of people.