I have the following: $$\int 0^1\int {\sqrt{4-y^2}}^{-1-\sqrt{1-y}}f(x,y)dxdy+\int _1^2\int _{\sqrt{4-y^2}}^0f(x,y)dxdy+\int _0^1\int _{-1+\sqrt{1-y}}^0f(x,y)dxdy::$$
How can I draw in the xy plane the integration domain? how is it possible when in the second and third integrals, the dx goes from the higher to the lower?!
One more thing, how can I change the order of integration? Instead of 3 integrals dxdy, I need to somehow get one integral dydx.
Thank for the helpers! ! !
#How can I change the order of integration, and how can I draw the integration domain?
vestal mist
olive spindleBOT
I have the following: $$\int _0^1\int_ {\sqrt{4-y^2}}^{-1-\sqrt{1-y}}f(x,y)dxdy+...
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turbid jungle
Try drawing all the boundary functions first.
subtle finchBOT
gingerGM
vestal mist
Try drawing all the boundary functions first.
I get something between the parabola $-x^2-2x$ and $\sqrt{4-y^2}$ but they are not continuous
subtle finchBOT
gingerGM
turbid jungle
I get something between the parabola $-x^2-2x$ and $\sqrt{4-y^2}$ but they are n...
Can you show your picture?
turbid jungle
Hm... Well, for the given functions, seems good.
Can you draw the regions for each integral?
vestal mist
@turbid jungle This is approximatly what I get
Im not sure about the green region
and about the blue one I can't "draw" it
because they are not connected to the $\sqrt{4-y^2}$
subtle finchBOT
gingerGM
turbid jungle
Im not sure about the green region
No, that's good.
What about the second integral? You'll probably need a separate picture for it, as it overlaps with the first region.
vestal mist
No, that's good.
What about the second integral? You'll probably need a separate...
Wait!
i DONT UNDERSTAND HOW IS IT GOOD
because I somhow start from $\sqrt{4-y^2}$ and end up in $0$ which is higher than it
turbid jungle
Why? Sure, the limits are swapped, but we can just swap them back and add a minus sign in front.
subtle finchBOT
gingerGM
vestal mist
Why? Sure, the limits are swapped, but we can just swap them back and add a minu...
yes, but then how do we add them together
or do you say we will deal with that problem later?
turbid jungle
Yup.
Hold on, I'll show how you can deal with the case of an opposite orientation.
turbid jungle
Like this.
turbid jungle
Now, we see that there is a portion where a "positive" region and a part of "negative" region overlap.
So, what happens there?
vestal mist
So, what happens there?
0?
turbid jungle
Yeah!
turbid jungle
So, we get this. The red region is now completely excluded.
So, now try getting the equations of these curves in the form y = f(x).
There are only two regions now.
turbid jungle
You flatter me 😅
You just need some practice.
vestal mist
And a better TO
turbid jungle
What's a TO?
vestal mist
$$\int_{-2}^{0}\int_{-x^2-2x}^{0}f(x,y)dydx+\int_{0}^{2}\int_{\sqrt{4-x^2}}^{0}f(x,y)dydx$$
Is that my answer?
turbid jungle
No, you didn't switch the order.
subtle finchBOT
gingerGM
turbid jungle
$$\int_{-2}^{0}\int_{-x^2-2x}^{0}f(x,y)dydx+\int_{0}^{2}\int_{\sqrt{4-x^2}}^{0}f...
Ah, that's better.
turbid jungle
Does the first 3 integrals add up to this?
Yes.
turbid jungle
But what does TO mean, exactly?
vestal mist
I meant tutor
turbid jungle
Oh, ok.