s(t) = 0.29t2 − t + 25 gallons per year (0 ≤ t ≤ 7), where t is time in years since the start of 2007. Use a Riemann sum with n = 5 to estimate the total country's per capita sales of bottled water from the start of 2008 to the start of 2012. (Use a left Riemann sum to find your answer. Round your answer to the nearest gallon.) HINT [See Example 2.]

To calculate that area, we divide it into a number of rectangles with one point touching the curve. The curve has a closed interval [a,b] that can be subdivided into n subintervals, each having a width of Δ[tex]x_{k}[/tex] = [tex]\frac{b-a}{n}[/tex]

If a function is defined on the closed interval [a,b] and [tex]c_{k}[/tex] is any point in [[tex]x_{k-1}[/tex],[tex]x_{k}[/tex]], then a Riemann Sum is defined as ∑f([tex]c_{k}[/tex])Δ[tex]x_{k}[/tex].

For this question:

Δ[tex]x_{k}[/tex] = [tex]\frac{7-0}{5}[/tex] = 1.4

Now, we have to find s(t) for each valor on the interval:

s(t) = 0.29[tex]t^{2}[/tex] - t +25

s(0) = 25

s(1) = 24.29

s(2) = 24.16

s(3) = 24.61

s(4) = 25.64

s(5) = 27.25

s(6) = 29.44

s(7) = 32.21

Now, using the formula:

∑f([tex]c_{k}[/tex])Δ[tex]x_{k}[/tex] = 1.4(25+24.29+24.16+24.61+25.64+29.44+32.21)

∑f([tex]c_{k}[/tex])Δ[tex]x_{k}[/tex] = 1.4(212.6)

∑f([tex]c_{k}[/tex])Δ[tex]x_{k}[/tex] ≅ 300

With Riemann Sum, it is estimated the total country's per capita sales of bottled water is 300 gallons.