Question

Suppose that the concentration (measured in gm^-3) of nitrogen in the soil along a transect in moist tundra yields data points that follow a straight line with equation y = 673.8 - 34.7x for 0 (is less than or greater to) x (is less than or greater to) 10, where x is the distance to the beginning of the transect.
What is the average concentration of nitrogen in the soil along this transect?

Answer 1

Answer:

The average concentration of nitrogen in the soil along  0 ≤ x ≤ 10 is $y_{avg}=500.3 \:{\frac{g}{m^3} }$

Step-by-step explanation:

To find the average concentration between two points a and b along a transect, we measure the concentration at equal distances. Assume that f(x) is a continuous function on [a, b]. The average value of f on the interval [a, b] is

$f_{avg}=\frac{1}{b-a} \int\limits^b_a {f(x)} \, dx$

To find the average concentration of nitrogen we have to take the integral over the entire range and then dividing by the distance covered in the study.

$y_{avg}=\frac{1}{10-0}\int\limits^{10}_{0} {(673.8 - 34.7x)} \, dx \\\\\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\y_{avg}=\frac{1}{10-0}\int _0^{10}673.8dx-\int _0^{10}34.7xdx\\\\y_{avg}=\frac{1}{10-0}(6738-1735)\\\\y_{avg}=\frac{5003}{10}=500.3$