(1 point) The following table gives the the approximate amount of emissions, $E$, of nitrogen oxides in millions of metric tons per year in the US. Let $t$ be the number of years since 1940 and $E=f(t)$. \begin{tabular}{|c|c|c|c|c|c|c|} \hline$t$ & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\ \hline$E$ & 6.6 & 9.8 & 13.2 & 18.2 & 21.3 & 19.5 \\ \hline \end{tabular} (a) Using left endpoints, estimate the integral: \[ \int_{0}^{50} f(t) d t \approx \] (b) Using right endpoints, estimate the integral: \[ \int_{0}^{50} f(t) d t \approx \] Be sure that you know what the units of your answer are, and what its meaning is! (Original data from the Statistical Abstract of the US, 1992)
Solution
Step 1 ：Using left endpoints, we estimate the integral by approximating the area under the curve as a series of rectangles. The width of each rectangle is the time interval (10 years), and the height is the emission at the start of each interval.
Step 2 ：The integral from 0 to 50 is approximately: \[\int_{0}^{50} f(t) d t \approx (10)(6.6) + (10)(9.8) + (10)(13.2) + (10)(18.2) + (10)(21.3)\]
Step 3 ：Calculating the above expression: \[\int_{0}^{50} f(t) d t \approx 660 + 980 + 1320 + 1820 + 2130 = 6910\]
Step 4 ：\(\boxed{\text{So, the approximate integral from 0 to 50 using left endpoints is 6910 million metric tons.}}\)
Step 5 ：Using right endpoints, we estimate the integral by approximating the area under the curve as a series of rectangles. The width of each rectangle is the time interval (10 years), and the height is the emission at the end of each interval.
Step 6 ：The integral from 0 to 50 is approximately: \[\int_{0}^{50} f(t) d t \approx (10)(9.8) + (10)(13.2) + (10)(18.2) + (10)(21.3) + (10)(19.5)\]
Step 7 ：Calculating the above expression: \[\int_{0}^{50} f(t) d t \approx 980 + 1320 + 1820 + 2130 + 1950 = 7200\]
Step 8 ：\(\boxed{\text{So, the approximate integral from 0 to 50 using right endpoints is 7200 million metric tons.}}\)
Step 9 ：The units of these answers are million metric tons, and they represent the total amount of nitrogen oxides emitted in the US from 1940 to 1990.
