Consider the closed and bounded set BC R² = [the zy-plane] defined byB = [0, 1] × [0, 1] = {(r, y) € R² : 0 ≤ x ≤ 1 and 0 ≤ y ≤l},and the closed and bounded set HCR²= [the ur-plane] defined byH ={(u,v) € R² : 0 ≤ u ≤ √2 and −min{v, √2-u} ≤ v ≤ min{x, √2-1)}.Draw a large picture of B and draw a large picture of H.Also consider the locally linear function T: R2 →R² given by(x, y) = T(u, v) := ( √ (u – v), √/21 (1 + r))v)√√2for all (u, v) € R².The function T rotates cach point (u, v) by π/4 radians, counterclockwise,around the origin (0,0). It is clear that T: Hº →Bº is one-to-one andonto, and T: OH OB is one-to-one and onto. Hence, T: HB isone-to-one and onto. You may assume these facts.Next we define the function f: BR by f(1, 1) := 0, andfor all (x, y) € B\{(1,1)} .1f(x, y) :=1-xy'(1) The function f is continuous at every point (x, y) € B, except (1,1).calculate1:= SS f(x,y) dA.(ry) € B(2) Calculate the integral1:=ſf_f(x,y) dA=(z.y) EBdirectly, by expanding•I=1x=0J:=1= (g(u, v), h(u, v)),y=01f(x, y):1. - xyas an infinite sum of non-negative polynomial functions gn:IMf(x,y) dy) af(x,y) = Σ9n(x, y), for every (x,y) € B\{(1, 1)}*;n=0dxand then using the fact that: (the integral of the infinite sum of these functions9n(1, y) over B] equals (the infinite sum of the integral of each gn(x, y) overB].(3) Using Parts (1) and (2) above, calculate exactly the infinite sum

note : Since you have posted multiple questions with multiple sub parts, we will provide the solution only to the first three sub parts of the first question as per Q&A guidelines. Please repost the remaining question by specifying the sub parts need to be answered. As per the question we are given two 2d regions B & H on the xy-plane and the corresponding linear transformation T : B → H Then we are given a bivariate function as : f(x,y) = 1/(1 - xy) for, (x,y) ∈ B\(1,1) And we have to : Plot the regions B & H find the double integral of f(x,y) over the domain B find the double integral of f(x,y) over the domain B using the infinite series expansion of f(x,y)Region B is clearly a unit square having vertices as (0,0) , (1,0) , (1,1) , (0,1) And as region H is just a 45o rotation of the region B , it is also a unit square having vertices as (0,0) , (1/√2,1/√2) , (√2,0) , (1/√2,-1/√2)