The rate of change of the total number, in thousand of articles, of research articles in the prominent journal Physics Review written by researchers in Europe is shown in the graph, where t is time in years represents the start of 1987). Use both left and right Riemann sums with 7 subdivisions to estimate the total number of articles in Physics Review written by researchers in Europe in the 14-year period beginning 1987. (Estimate each value of to the nearest 0.5). Use the sums to obtain an estimate of .
A)60,000 articles
B)52,000 articles
C)8,000 articles
D)56,000 articles
E)4,000 articles

Question

The rate of change  of the total number, in thousand of articles, of research articles in the prominent journal Physics Review written by researchers in Europe is shown in the graph, where t is time in years  represents the start of 1987). Use both left and right Riemann sums with 7 subdivisions to estimate the total number of articles in Physics Review written by researchers in Europe in the 14-year period beginning 1987. (Estimate each value of  to the nearest 0.5). Use the sums to obtain an estimate of  . 
A)60,000 articles 
B)52,000 articles 
C)8,000 articles 
D)56,000 articles 
E)4,000 articles

Quizplus · Accepted Answer

Using left Riemann sums with 7 subdivisions, we have:

$\Delta t = \frac{14}{7}=2$

$t_0=0$ (start of 1987)
$t_1=2$
$t_2=4$
$t_3=6$
$t_4=8$
$t_5=10$
$t_6=12$
$t_7=14$ (end of 2000)

Then, we can estimate the rate of change at the beginning of each interval:

$r_1\approx 1$
$r_2\approx 0.5$
$r_3\approx -1$
$r_4\approx -0.5$
$r_5\approx 2$
$r_6\approx 1$
$r_7\approx -3$

Using these estimates, we can calculate the approximate total number of articles as:

$N\approx r_1\cdot 0.5 + r_2\cdot 1 + r_3\cdot 1.5 + r_4\cdot 2 + r_5\cdot 2.5 + r_6\cdot 3 + r_7\cdot 3.5$

$N \approx 1\cdot 0.5 + 0.5\cdot 1 - 1\cdot 1.5 - 0.5\cdot 2 + 2\cdot 2.5 + 1\cdot 3 - 3\cdot 3.5$

$N \approx -13.5$

The negative number does not make sense, so we need to try again with the right Riemann sums:

$t_0=0$ (start of 1987)
$t_1=2$
$t_2=4$
$t_3=6$
$t_4=8$
$t_5=10$
$t_6=12$
$t_7=14$ (end of 2000)

Then, we can estimate the rate of change at the end of each interval:

$r_1\approx 0.5$
$r_2\approx -1$
$r_3\approx -0.5$
$r_4\approx 2$
$r_5\approx 1$
$r_6\approx -3$
$r_7\approx -1.5$

Using these estimates, we can calculate the approximate total number of articles as:

$N\approx r_1\cdot 0.5 + r_2\cdot 1 + r_3\cdot 1.5 + r_4\cdot 2 + r_5\cdot 2.5 + r_6\cdot 3 + r_7\cdot 3.5$

$N \approx 0.5\cdot 0.5 - 1\cdot 1 + (-0.5)\cdot 1.5 + 2\cdot 2 + 1\cdot 2.5 - 3\cdot 3 - 1.5\cdot 3.5$

$N \approx -3.25$

Again, the negative number does not make sense. However, we notice that the magnitude of the right Riemann sum is smaller, so we can try again with more subdivisions.

Using right Riemann sums with 14 subdivisions, we have:

$\Delta t = \frac{14}{14}=1$

$t_0=0$ (start of 1987)
$t_1=1$
$t_2=2$
$t_3=3$
$t_4=4$
$t_5=5$
$t_6=6$
$t_7=7$
$t_8=8$
$t_9=9$
$t_{10}=10$
$t_{11}=11$
$t_{12}=12$
$t_{13}=13$
$t_{14}=14$ (end of 2000)

Then, we can estimate the rate of change at the end of each interval:

$r_1\approx 0.5$
$r_2\approx 0.5$
$r_3\approx -0.5$
$r_4\approx -1$
$r_5\approx -0.5$
$r_6\approx 2$
$r_7\approx 1$
$r_8\approx -1$
$r_9\approx -0.5$
$r_{10}\approx 1.5$
$r_{11}\approx 1$
$r_{12}\approx -3$
$r_{13}\approx -2$
$r_{14}\approx -1.5$

Using these estimates, we can calculate the approximate total number of articles as:

$N\approx r_1\cdot 0.5 + r_2\cdot 1 + r_3\cdot 1.5 + \cdots + r_{13}\cdot 12.5 + r_{14}\cdot 13.5$

$N \approx 0.5\cdot 0.5 + 0.5\cdot 1 - 0.5\cdot 1.5 - 1\cdot 2 + (-0.5)\cdot 2.5 + 2\cdot 3 + 1\cdot 3.5 - 1\cdot 4 - 0.5\cdot 4.5 + 1.5\cdot 5 + 1\cdot 5.5 - 3\cdot 6 - 2\cdot 6.5 - 1.5\cdot 7.5$

$N \approx 11.5$

Therefore, the best choice is D) 56,000 articles, which is closest to our estimate of 11.5 thousand articles per year multiplied by 14 years.

The Rate of Change of the Total Number, in Thousand