How do I find the volume under a circular domain using double integrals?

[Shaybay92]

So I have to use the type I type II region formula to find the volume under the equation (2x-y) and over the circular domain with center (0,0) and radius 2. Do I have to split this circle into hemispheres and treat it as 2 type I domains? I got the following limits for the top half, but I get stuck when integrating:

y limits:
Upper: Sqrt(2 - x^2) from the equation 2 = y^2 + x^2
Lower: 0

X limits:
Upper: 2
Lower: -2

So I have to find the integral with respect to y of 2x-y with limits 0 to Sqrt[2-x^2]

After integrating with respect to Y I got:

2x(Sqrt[2-x^2]) - 1 + (x^2)/2

Is this correct to start with? Then integrate with respect to x from -2 to 2?

 

[tiny-tim]

Hi Shaybay92!

(have a square-root: √ and an integral: ∫ and try using the X² tag just above the Reply box )

Can you please clarify the question?

Are you trying to find a 2D area, or a 3D volume?

By "the equation (2x-y)" do you mean the line (or plane) with 2x-y = 0?

If so, isn't this just the area of a semi-circle (or the volume of a hemisphere)?

(I'm not familiar with the "type and I type II" classification, but it looks like you need to use the area or volume under 2x-y=0 separately from the area or volume of the cap that's "clear" of 2x-y=0)