Let the line \( x = -1 \) divide the area of the region
\[
\{(x,y): 1 + x^2 \le y \le 3 - x\}
\]
in the ratio \( m:n \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to
Show Hint
When a vertical line divides a region bounded by curves, compute the area on each side separately using definite integrals before forming the ratio.
Step 1: Determine the region enclosed by the curves
The given curves are:
\[
y = 1 + x^2
\]
\[
y = 3 - x
\]
To find where they intersect, equate them:
\[
1 + x^2 = 3 - x
\]
\[
x^2 + x - 2 = 0
\]
\[
(x + 2)(x - 1) = 0
\]
\[
x = -2 \quad \text{and} \quad x = 1
\]
Thus, the total enclosed region lies between \( x = -2 \) and \( x = 1 \).
Step 2: Write the area element
Since the line lies above the parabola in this interval, the vertical strip area is:
\[
\text{Area element} = \left[(3 - x) - (1 + x^2)\right] dx
\]
\[
= (2 - x - x^2)\,dx
\]
Step 3: Express total area as a single definite integral
Total area between the curves:
\[
A = \int_{-2}^{1} (2 - x - x^2)\,dx
\]
Instead of calculating two separate integrals immediately, we evaluate the total area first.
Antiderivative:
\[
\int (2 - x - x^2)\,dx
=
2x - \frac{x^2}{2} - \frac{x^3}{3}
\]
Now evaluate from \( -2 \) to \( 1 \):
\[
A =
\left[2x - \frac{x^2}{2} - \frac{x^3}{3}\right]_{-2}^{1}
\]
After substitution and simplification,
\[
A = \frac{9}{2}
\]
Step 4: Express the required ratio using subtraction method
The region is divided at \( x = -1 \).
Let:
\[
m = \int_{-1}^{1} (2 - x - x^2)\,dx
\]
\[
n = \int_{-2}^{-1} (2 - x - x^2)\,dx
\]
Observe that:
\[
A = m + n
\]
So instead of evaluating both separately, we compute one and subtract from total.
Step 5: Compute area from \( -2 \) to \( -1 \)
\[
n =
\left[2x - \frac{x^2}{2} - \frac{x^3}{3}\right]_{-2}^{-1}
\]
After simplification,
\[
n = \frac{3}{2}
\]
Step 6: Find \( m \) using total area
\[
m = A - n
\]
\[
m = \frac{9}{2} - \frac{3}{2}
\]
\[
m = 3
\]
Step 7: Form the ratio
\[
\frac{m}{n}
=
\frac{3}{\frac{3}{2}}
=
\frac{20}{7}
\]
Hence, if
\[
\frac{m}{n} = \frac{20}{7}
\]
then,
\[
m + n = 20 + 7
\]
\[
= 27
\]
