How do I find the volume using cylindrical coordinates for the given region?

[Redoctober]

I need some help in this

The question states find the volume for the region bounded below by z = 3 - 2y and above by z = x^2 + y^2 using cylindrical coordinates .

Now i tried to do it but i got stuck with the part where i find the radius . i found
r^2 = 3-2r*sin(zeta) :( its hard to factorize and use this for the boundary

i already know that z will range from r^2 to 3-2r*sin(zeta) and zeta range from zero to 2 pie

so how can this question be solved ? :)

btw the correct answer is 8pie

 

[HallsofIvy]

First, you have "below" and "above" reversed. The plane z= 3- 2y is above the paraboloid z= x^2+ y^2 in the bounded region.

They intersect where z= x^2+ y^2= 3- 3y which is equivalent to x^2+ y^2+ 3y= 3. x^2+ y^2+ 3y+ 9/4= x^2+ (y+3/2)^2= 3+ 9/4= 21/4. That is a circle with center at (0, -3/2) which is why ordinary polar coordinates do not give a simple equation. Either
1: integrate with x from -\sqrt{21}/2 to \sqrt{21}/2 and, for each x, y from -3/2- \sqrt{21/4- x^2} to -3/2+ \sqrt{21/4- x^2}.

2: Let x= rcos(\theta) and y= -3/2+ rsin(\theta), shifting the origin to (0, -3/2).

 

[Redoctober]

Oh i c :D ! Thanks a lot . I wasn't aware of shifting the origin :) !

Thanks :D !

 

[redrum419_7]

LOL 8pie

 

[HallsofIvy]

For desert?