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# In a report issued by the Economist Intelligence Unit in 2013, the number of commercial vehicle registrations in the United States (in thousands) by year from 2010 through 2015 is approximated by the function $N(t)=1.3926 t^{3}-9.2873 t^{2}+74.719 t+228.3$ $(0 \leq t \leq 5)$ where t is measured in years with t = 0 corresponding to 2010. If the projection holds up, what would the approximate average commercial vehicle registration per year be in the period from 2010 through 2015?

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The average commercial vehicle registration per year between the year 2010 and 2015 would be given by

\begin{align*} \frac{1}{5 - 0} \int_0^{5} N(t)\ dt & = \frac{1}{5} \int_0^{5} (1.3926t^3 - 9.2873t^2 + 74.719t + 228.3)\ dt\\ & = \frac{1}{5} \bigg(0.34815t^4 - \frac{9.2873}{3}t^3 + 37.3595t^2 + 228.3t\bigg)\bigg|_0^{5}\\ & = \frac{1}{5} \bigg[ \bigg(0.34815(5)^4 - \frac{9.2873}{3}(5)^3 + 37.3595(5)^2 + 228.3(5)\bigg)\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \bigg(0.34815(0)^4 - \frac{9.2873}{3}(0)^3 + 37.3595(0)^2 + 228.3(0)\bigg) \bigg]\\ & = \frac{1}{5} [ 1906.11 - 0]\\ & = 381,222 \end{align*}

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