Let the line $x = - 1$ divide the area of the region $\left\{ (x,y):1 + x^{2} \leq y \leq 3 - x \rig

Question

Let the line $x = - 1$ divide the area of the region $\left\{ (x,y):1 + x^{2} \leq y \leq 3 - x \right\}$ in the ratio $m:n,gcd(m,n) = 1$. Then $m + n$ is equal to

Accepted Answer

$${\frac{m}{n} = \frac{\int_{- 1}^{1}\mspace{2mu}\left\lbrack (3 - x) - \left( 1 + x^{2} \right) \right\rbrack dx}{\int_{- 2}^{1}\mspace{2mu}\left\lbrack (3 - x) - \left( 1 + x^{2} \right) \right\rbrack dx} = \frac{20}{7} }{\therefore m + n = 20 + 7 }{= 27}$$

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