Change the order of integration of `int_0^1int_(-sqrt(2y-y^2))^(1+sqrt(1-y^2)) f(x,y)dxdy`
Let `int_0^1int_(-sqrt(2y-y^2))^(1+sqrt(1-y^2)) f(x,y)dxdy`
Region of integration : `-sqrt(2y-y^2)`≤ 𝒙 ≤ 𝟏+`sqrt(1-y^2)`
Curves : (i) `x=-sqrt(2y-y^2) => x^2+y^2=2y =>x^2+(y-1)^2=1`
Circle with centre (0,1) and radius 1.
(ii) `x=1+sqrt(1-y^2) => (x-1)^2+y^2=1`
Circle with centre (1,0) and radius 1.
(iii) 𝒚=𝟎 𝒍𝒊𝒏𝒆 𝒊.𝒆 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒙−𝒂𝒙𝒊𝒔.
(iv) 𝒚=𝟏 𝒍𝒊𝒏𝒆 𝒑𝒂𝒓𝒂𝒍𝒍𝒆𝒍 𝒕𝒐 𝒙−𝒂𝒙𝒊𝒔.
Divide the region R into R1 and R2
∴ R = R1 ∪ R2
After changing the order of integration ,
For region R1 : 𝟎≤𝒚≤𝟏−`sqrt(1-x^2)`
For region R2 : 0 ≤𝒚≤ `sqrt(1-(x-1)^2)`
𝟏 ≤ 𝒙 ≤ 𝟐
As the region is divided in two parts the integration will be the union of the two region limits.
This is the integration after changing order from dx dy to dy dx of given integration region.