Question

If the area of the region \( \{ (x, y) : x^2 + 1 \leq y \leq 3 - x \} \) is divided by the line \( x = -1 \) in the ratio \( m : n \) (where \( m \) and \( n \) are coprime natural numbers), then the value of \( m + n \) is:

Hint:

When dividing areas using a line, compute the area on either side of the line and find the ratio of these areas.

Answer 1

Correct Answer: 27

Step 1: Finding the total area of the region.
The given region is bounded by the curves \( y = x^2 + 1 \) and \( y = 3 - x \). To find the area of this region, we compute the integral of the difference between the two functions over the range of \( x \). Step 2: Calculating the area divided by the line \( x = -1 \).
The area to the left of \( x = -1 \) is calculated by integrating over the appropriate bounds, and similarly for the area to the right. The ratio \( m : n \) is obtained from the two areas. Step 3: Conclusion.
The total area is divided in the ratio \( m : n = 16 : 11 \), and thus \( m + n = 27 \). Final Answer: \[ \boxed{27} \]