the following table gives the annual natural gas production, in trillions of cubic feet per year, in the us…

the following table gives the annual natural gas production, in trillions of cubic feet per year, in the us between 2002 and 20141. year 2002 2004 2006 2008 2010 2012 2014 gas (tr ft³/yr) 19.9 19.5 19.4 21.1 22.4 25.3 27.5 estimate the total natural gas produced in the us between 2002 and 2014 using a: (i) left sum, n = 6 : estimate = trillions of cubic feet. (ii) right sum, n = 6 : estimate = trillions of cubic feet.

the following table gives the annual natural gas production, in trillions of cubic feet per year, in the us between 2002 and 20141. year 2002 2004 2006 2008 2010 2012 2014 gas (tr ft³/yr) 19.9 19.5 19.4 21.1 22.4 25.3 27.5 estimate the total natural gas produced in the us between 2002 and 2014 using a: (i) left sum, n = 6 : estimate = trillions of cubic feet. (ii) right sum, n = 6 : estimate = trillions of cubic feet.

Answer

Explanation:

Step1: Determine the time - step

The time interval is from 2002 to 2014, so $\Delta t=\frac{2014 - 2002}{6}=2$ years.

Step2: Calculate the left - sum

The left - sum formula for a sum $L_n=\sum_{i = 0}^{n - 1}f(x_i)\Delta x$. Here, $x_i$ are the left - hand endpoints of the sub - intervals and $f(x)$ is the natural gas production. The left - hand values for $n = 6$ sub - intervals are: $f(2002)=19.9$, $f(2004)=19.5$, $f(2006)=19.4$, $f(2008)=21.1$, $f(2010)=22.4$, $f(2012)=25.3$. Then $L_6=(19.9 + 19.5+19.4 + 21.1+22.4+25.3)\times2=(127.6)\times2 = 255.2$.

Step3: Calculate the right - sum

The right - sum formula for a sum $R_n=\sum_{i = 1}^{n}f(x_i)\Delta x$. The right - hand values for $n = 6$ sub - intervals are: $f(2004)=19.5$, $f(2006)=19.4$, $f(2008)=21.1$, $f(2010)=22.4$, $f(2012)=25.3$, $f(2014)=27.5$. Then $R_6=(19.5 + 19.4+21.1+22.4+25.3+27.5)\times2=(135.2)\times2=270.4$.

Answer:

(i) 255.2 (ii) 270.4