Annual U.S. imports from China in the years 1997-1999 could be approximated by $I = t ^ { 2 } + 3.5 t + 50 ( 2 \leq t \leq 4 )$ billion dollars per year, where t represents time in years since the start of 1995. During the same period, annual U.S. exports to China could be approximated by $E = 0.4 t ^ { 2 } - 1.6 t + 14 ( 2 \leq t \leq 4 )$ billion dollars per year.
Compute the area between the graphs of I and E over the interval $[ 2,4 ]$ . (Round your answer to the nearest $10 billion.)

A)$80 billion
B) $170 billion
C) $230 billion
D) $20 billion
E) $110 billion

Question

Annual U.S. imports from China in the years 1997-1999 could be approximated by $I = t ^ { 2 } + 3.5 t + 50 ( 2 \leq t \leq 4 )$ billion dollars per year, where t represents time in years since the start of 1995. During the same period, annual U.S. exports to China could be approximated by $E = 0.4 t ^ { 2 } - 1.6 t + 14 ( 2 \leq t \leq 4 )$ billion dollars per year.  
Compute the area between the graphs of I and E over the interval $[ 2,4 ]$ . (Round your answer to the nearest $10 billion.)
 
A)$80 billion
B) $170 billion
C) $230 billion
D) $20 billion
E) $110 billion

Quizplus · Accepted Answer

To find the area between the graphs of $I$ and $E$ over the interval $[2,4]$, we need to integrate the difference between the two functions over this interval. The difference $I - E$ is given by:$$\begin{align*}I - E &= (t^2 + 3.5t + 50) - (0.4t^2 - 1.6t + 14) \&= t^2 - 0.4t^2 + 3.5t + 1.6t + 50 - 14 \&= 0.6t^2 + 5.1t + 36.\end{align*}$$Integrating this expression from $t = 2$ to $t = 4$ gives the area between the graphs:$$\begin{align*}\int_{2}^{4} (0.6t^2 + 5.1t + 36) dt &= \left[0.2t^3 + \frac{5.1}{2}t^2 + 36t\right]_{2}^{4} \&= \left(0.2(4)^3 + \frac{5.1}{2}(4)^2 + 36(4)\right) - \left(0.2(2)^3 + \frac{5.1}{2}(2)^2 + 36(2)\right) \&= \left(0.2(64) + \frac{5.1}{2}(16) + 144\right) - \left(0.2(8) + \frac{5.1}{2}(4) + 72\right) \&= \left(12.8 + 40.8 + 144\right) - \left(1.6 + 10.2 + 72\right) \&= 197.6 - 83.8 \&= 113.8.\end{align*}$$Rounding to the nearest $10 billion, the area between the graphs over the interval $[2,4]$ is approximately $110 billion.

Annual US Imports from China in the Years 1997-1999 Could Be