Application of Integrals

Let the area of the region \(x, y): x-2y+4 0, x+2y2 0, x+4y2 8, y 0 \ be mn, where m and n are coprime numbers. Then m+n is equal to .

Answer

119

Solution

[figure]

Left boundary: x=-2y2 for 0 y 1, x=2y-4 for y 1 (they meet at (-2, 1))

Right boundary: x=8-4y2; it meets x=2y-4 at 2y2+y-6=0 ⇒ y= 32

Area = 01 [(8-4y2)-(-2y2) ]dy+ 13/2 [(8-4y2)-(2y-4) ]dy

= [8y- 2y33 ]01+ [12y-y2- 4y33 ]13/2

= 223+ ( 454- 293 )= 223+ 1912= 10712

∴ m+n=107+12=119

Answer: 119