Let the area of the region ( x , y ) : x - 2 y + 4 ≥ 0 , x + 2 y 2 ≥ 0 , x + 4 y 2 ≤ 8 , y ≥ 0 be m n , where m and n are coprime numbers. Then m + n is equal to ______.

Step-by-step solution

Given, x - 2 y + 4 ≥ 0 , x + 2 y 2 ≥ 0 , x + 4 y 2 ≤ 8 , y ≥ 0 Now, finding the point of intersection of x - 2 y + 4 = 0 & x + 2 y 2 = 0 we get, x , y ≡ - 2 , 1 And point of intersection of x - 2 y + 4 = 0 & x + 4 y 2 = 8 will be, x , y ≡ - 1 , 3 2 Now, plotting the diagramw we get, Now, from the above diagram, the required area will be, A = ∫ - 2 - 1 x + 4 2 - - x 2 + ∫ - 1 0 8 - x 2 - - x 2 d x + ∫ 0 8 8 - x 2 d x ⇒ A = 1 2 x + 4 2 2 - 2 - 1 - 1 2 - x 3 2 - 2 3 - 2 - 1 + 1 2 8 - x 3 2 - 2 3 - 1 0 - 1 2 - x 3 2 -