What are the boundaries for a double integral over a specific region?

[Kuma]

Homework Statement

D = {x>0, x^2 < y < 10-x^2)

compute

integral (integral D of y^2 sqrt x)

Homework Equations

The Attempt at a Solution

I'm having trouble figuring out the bounds of the integral. y goes from x^2 to 10-x^2 but I think I have to split this integral up into two parts. I am not sure how to bound x. The parabolas intersect at the point sqrt 5

 

[flyingpig]

What do you mean probably at sqrt(5)? They intersect at two points

 

[Kuma]

They intersect at (sqrt 5, 5) and (-sqrt 5, 5). x > 0 so we don't need the negative point.

 

[flyingpig]

Yes you got, it

 

[Kuma]

So x goes from 0 to sqrt 5? that's the bound for x?

 

[flyingpig]

So x goes from 0 to sqrt 5? that's the bound for x?

What do you think it should be? Let's put it this way, if it isn't x = 0, where would you start from? You mentioned splitting the integral, how do you plan to do that?

 

[Kuma]

I guess I'm overthinking. It looks like x goes from 0 to x^2 and then stops when x reaches sqrt 5, then goes from 10-x^2 back to 0. This is just from the drawing I mean.

 

[Redbelly98]

So x goes from 0 to sqrt 5? that's the bound for x?

Yes, provided that you integrate over y first -- which, as you said, goes from x^2 to 10-x^2.