Final answer:
The area of region R bounded by f(x) = 2x and g(x) = -x + 97 is found by integrating the difference of the functions from 0 to 97/3.
Step-by-step explanation:
To find the area A of the region R bounded by the functions f(x) = 2x and g(x) = -x + 97, we need to set up an integral where we subtract one function from the other over the interval where they intersect. First, we find the points of intersection by equating f(x) and g(x):
2x = -x + 97 \Rightarrow 3x = 97 \Rightarrow x = 97/3.
Since these are lines, they will only intersect at one point and continue on in opposite directions. Therefore, our integration limits will be from 0 to 97/3, and the integral is:
\int_{0}^{97/3} (2x - (-x + 97)) dx = \int_{0}^{97/3} (3x - 97) dx.
To solve this integral, we split it up:
\int_{0}^{97/3} 3x dx - \int_{0}^{97/3} 97 dx.
The solution to the integral gives the area of region R which is the absolute value of the difference between the areas under f(x) and g(x).