I am having some trouble with the following 2 questions on improper multiple integrals. I hope that someone can help me out! 1) Determine whether I=∫∫ cos(sqrt(x2+y2)) / (x2+y2) converges or diverges. x,y>1 Solution: Let R=[0,1]x[0,1] B(0,1)=ball of radius 1 centered at origin R+xR+=positive xy-plane f(x,y)=the integrand The solution says that I=∫∫ f(x,y) dA - ∫∫ f(x,y) dA R+xR+ \B(0,1) R\B(0,1) And then showed that both integrals converge, so the given improper integral I converges. ============================ Now, I am having a lot of trouble understanding the red part, WHY is it true? ============================ 2) Determine whether the following converges or diverges. Let S=[-1,1]x[-1,1] 2a) ∫∫ x2 / (x2+y2) dA S 2b) ∫∫ sqrt|x| / (x2+y2) dA S ============================ In the solutions manual, seemingly, they know the answer at the beginning; they inscribed a circle within the rectangle S for 2a) and inscribed the rectangle S in a circle for 2b), said that the integrand >0 except the origin, and used the comparsion test to conclude the first one diverges and the second converges. ============================ Now, I just want to know HOW I can get a first feeling about whether the above improper integrals will converge or diverge before going into the details. It's nice to know the answer ahead of time, so that I can know which direction to push forward the proof. Otherwise, I will just be doubling my amount of time and effort to finish. Thank you for explaining!