Question

Hi, can someone please help me with this. I really appreciate it.

Given the sum of two iterated integrals
I = 0∫(√2)/2 0 ∫x √(x2 + y2) dydx + (√2)/2 ∫1 0 ∫√1-x^2) √(x2 + y2) dydx

a) use the set notation to describe the region D of integration

b) sketch region D

I think I have a general idea, but I'm unsure. Do I set the intervals of dy and dx equal to each other?
Thanks again.

Answer 1

Don't understand your notation: what is the 0s in front of integral sign?

Answer 2

My bad. I'll rewrite it.

Answer 3

The first one seems to be describing the line y=x

Answer 4

The whole region is a semicircle of radius one, pierced by y=x

Answer 5

Thanks, but i"m still lost.
What would the interval of the region be?

Answer 6

First one 0<y<x
0<x<(sq rt 2)/2

Answer 7

Second one 0<y<sq rt (1-x^2)
(sq rt2)/2<x<1

Answer 8

Oh, so I don't have to write a notation that combines the two iterated integrals as one?

Answer 9

I was just 'quoting' what the integrals say to answer your question, I wasn't working on the problem so to speak.

Answer 10

The first one describes a region that looks like a piece of pie between 0 and y=x of length (sq rt 2)/2. And the second part goes from the end of that starting at (sq rt 2)/2 to 1 a sector of a semicircle.

Answer 11

Okay, thanks a lot!