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

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

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)=(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)=(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)=(1)/(10-0)\int _0^(10)673.8dx-\int _0^(10)34.7xdx\\\\y_(avg)=(1)/(10-0)(6738-1735)\\\\y_(avg)=(5003)/(10)=500.3`