To find the area of the region enclosed by the curves [tex]\( y = x \)[/tex] and [tex]\( y = \frac{5.75x}{x^2 + 1} \)[/tex], we need to calculate the integral of the absolute difference between these two functions over the given range.
Here's the step-by-step solution:
1. Understand the functions to be integrated:
- The first function is [tex]\( f_1(x) = x \)[/tex].
- The second function is [tex]\( f_2(x) = \frac{5.75x}{x^2 + 1} \)[/tex].
2. Determine the absolute difference between the two functions:
The absolute difference between the functions [tex]\( f_1(x) \)[/tex] and [tex]\( f_2(x) \)[/tex] is:
[tex]\[
|f_1(x) - f_2(x)| = \left| x - \frac{5.75x}{x^2 + 1} \right|
\][/tex]
3. Set up the integral over the given range:
To find the area enclosed by these two curves, we need to integrate this absolute difference over the chosen range from [tex]\( x = -10 \)[/tex] to [tex]\( x = 10 \)[/tex].
4. Compute the integral:
The integral we need to solve is:
[tex]\[
\int_{-10}^{10} \left| x - \frac{5.75x}{x^2 + 1} \right| \, dx
\][/tex]
5. Evaluate the integral:
After evaluating this integral, the result is found to be:
[tex]\[
\int_{-10}^{10} \left| x - \frac{5.75x}{x^2 + 1} \right| \, dx = 84.0788553430287
\][/tex]
So, the area of the region enclosed by the curves [tex]\( y = x \)[/tex] and [tex]\( y = \frac{5.75x}{x^2 + 1} \)[/tex] is [tex]\( \boxed{84.0788553430287} \)[/tex].
