Iterated integral clarification

[Derill03]

There is an iterated integral:

outer integration = 1 to -1 (limits of int.)
inner integration = sqrt(1-x^2) to -sqrt(1-x^2)
function = x^2 + y^2

the part i don't get is the question states to sketch or describe the region, I am good with that, but i don't understand what is meant by the following:

What region in the plane comprises the lower boundary of this region?

Can anyone help?

 

[tiny-tim]

… i don't understand what is meant by the following:

What region in the plane comprises the lower boundary of this region?

Hi Derill03!

I agree it's a strange question …

the boundary of a region in the plane is a curve, not a region …

I suppose it's asking for the bottom half of the circle.

 

[s10dude04]

Hello Derill03

I believe I am faced with the same question as you.

My question is what the graph of this would look like in R3. Is it an inverted conical shape or a sphere?

The lower boundary that TinyTim was mentioning, "lower half of the circle", is this referring a lower half of a sphere?

 

[Mark44]

Hello Derill03

I believe I am faced with the same question as you.

My question is what the graph of this would look like in R3. Is it an inverted conical shape or a sphere?

The lower boundary that TinyTim was mentioning, "lower half of the circle", is this referring a lower half of a sphere?

No, it's the lower half of a circle in the x-y plane. We're talking about the region over which (double) integration takes place, so this region is two-dimensional. OTOH, the integral itself might represent the volume of a 3D object whose z-value is x^2 + y^2, which is a cone.

Derill03,
You listed the limits of integration as

outer integration = 1 to -1 (limits of int.)
inner integration = sqrt(1-x^2) to -sqrt(1-x^2)

The usual practice is to list the lower limit first, and then the upper limit, not the other way round, as you seem to have done.