Chapter 14.1, Problem 59E

Housing The following graph shows the annual number of housing starts in the United States during 2000–2008 together with a quadratic approximating model:

$s\left(t\right)=-30t^2\text{+240}t+800\text{ thousand homes per year}\left(0\le t\le 8\right)$

(t is the time in years since 2000.)⁴ At the same time, the homes being built were getting larger:

$a\left(t\right)=\text{40}t+2,000\text{ square feet}\text{.}$

Use the given models to estimate the total housing area under construction over the given period. (Use integration by parts to evaluate the integral, and round your answer to the nearest billion square feet.) [HINT: Rate of change of area under construction $=p\left(t\right)q\left(t\right)$.]