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kingwinner
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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!
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!
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