Part I
The change in annual revenue in thousands of dollars for the Ozark Zip Line Corporation has been modeled by the function.
R(x) = 8.2x^2 + 23.1x
where x is the number of years after 2000. To find the total revenue gained between the years 2002 and 2005, start by choosing a constant (C)530
Part II
Build the definite integral that is used to find the total revenue gained between the years 2002 and 2005
Part III
Apply the power rule to integrate R(x) over the given interval, and round your answer to the nearest whole tens of dollars.
Part I:
To find the total revenue gained between the years 2002 and 2005, we need to calculate the definite integral of the revenue function R(x) over this interval. Let's first determine the limits of integration:
- The year 2002 corresponds to x = 2, as it is 2 years after 2000.
- The year 2005 corresponds to x = 5, as it is 5 years after 2000.
Part II:
The definite integral to find the total revenue gained between the years 2002 and 2005 can be written as:
∫[2,5] R(x) dx
This integral represents the area under the curve of the revenue function R(x) from x = 2 to x = 5.
Part III:
To solve the definite integral, we need to apply the power rule for integration. The power rule states that if a function is of the form ax^n, then its integral with respect to x is (a/(n+1))x^(n+1) + C.
Let's apply the power rule to R(x) = 8.2x^2 + 23.1x:
∫(8.2x^2 + 23.1x) dx = (8.2/3)x^3 + (23.1/2)x^2 + C
Now, we can evaluate the definite integral over the interval [2,5]:
∫[2,5] R(x) dx = [(8.2/3)(5^3) + (23.1/2)(5^2)] - [(8.2/3)(2^3) + (23.1/2)(2^2)]
Calculating this expression will give us the total revenue gained between the years 2002 and 2005, rounded to the nearest whole tens of dollars.