I am having some trouble with the following 2 questions on improper multiple integrals. I hope that someone can help me out!(adsbygoogle = window.adsbygoogle || []).push({});

1) Determine whether

I=∫∫ cos(sqrt(x^{2}+y^{2})) / (x^{2}+y^{2}) 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.

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Now, I am having a lot of trouble understanding the red part, WHY is it true?

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2) Determine whether the following converges or diverges.

Let S=[-1,1]x[-1,1]

2a)

∫∫ x^{2}/ (x^{2}+y^{2}) dA

S

2b)

∫∫ sqrt|x| / (x^{2}+y^{2}) dA

S

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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.

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Now, I just want to know HOW I can get afirst feelingabout 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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# Homework Help: Improper Multiple Integrals

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