A sketch of the region enclosed by y=x and y=7.5x/x^2+1 follows.

The sketch above shows the region enclosed by the curves \( y = x \) (blue line) and \( y = \frac{7.5x}{x^2 + 1} \) (orange curve). The shaded gray area represents the region between these two curves from \( x \approx -2.55 \) to \( x \approx 2.55 \).

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### Related Questions:

1. How do you calculate the area of the region enclosed between these two curves?
2. What is the maximum value of the function \( y = \frac{7.5x}{x^2 + 1} \) and at what point does it occur?
3. How do the asymptotic behaviors of these functions influence the enclosed area?
4. What happens if we change the coefficient in the function \( y = \frac{7.5x}{x^2 + 1} \)?
5. Can the method used here be applied to find the region enclosed by other types of curves?

**Tip:** Understanding the intersection points of functions is crucial in finding enclosed areas and solving integration problems.

A sketch of the region enclosed by y=x and y=7.5x/(x^2+1) follows.

Explore how to find the area enclosed by the curves y = x and y = 7.5x/(x^2 + 1) using integration. Learn key calculus concepts such as curve sketching, finding intersection points, and calculating the area between two curves. Suitable for college-level calculus students.

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