Answer:
a. 20520
b. 12600
Step-by-step explanation:
Given the function S = 600xe^-x² + 600.
a. To find the average, we have to find the definite integral of the function because average, as it is known, is the sum of data points divided by the size of its dataset, this can be used for discrete data. Integral formula is just the continuous form of average, so we are using integral because we were given an interval of x= 0 to X = 30.
Let's integrate 100xe^x² first. Let –x² = u, this means –2xdx = du and we have dx = –du/2x. Also, when x = 0, u = –(0)² = –0 and when x = 30, u = –(30)² = –900. When we make our substitutions we have:
–600(xe^udu)/2x = –600(e^udu)/2 upon integrating that we have –600(e^u)/2. Applying our interval we have
–600[(e^900)/2 – (e^0)/2] ≈ – [– 3.7 – (1/2)] = –600 (–4.2) = 600 x 4.2 = 2520
Now let's integrate 600, with the interval x = 0 to x = 30 (we are using this interval here because the substitution we made didn't affect this).
We have, upon integrating:
600x.
Substituting our intervals we have:
600(30 – 0) = 600 x 30 = 18000.
Adding that up we have: 2520 + 18000 = 20520.
b. The same method is needed, just difference of interval.
Therefore, after integrating the first component with intervals u = 900 to u = 2500 (from x² = u) we have:
–600[(e^2500)/2 – (e^900)/2] ≈ –600[2.7 – 3.7] = –600(–1) = 600.
Then for the second component:
600x using x = 30 to x = 50
600(50 – 30) = 600 x 20 = 12000.
Adding that up we have:
12000 + 600 = 12600.
. The polynomial remainder theorem guarantees that the value of 
is the remainder upon dividing 
by 
, but I digress...
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