Multivariable Cal, (Polar Coordinates)

[wildleaf]

Homework Statement

Evaluate by changining to polar coordinates :
#19 in this picture http://i52.tinypic.com/2ngbt5z.jpg

Homework Equations

x = cos θ
y = sin θ

r^2 = x^2 + y^2 + z^2

∫∫∫w f(x,y) dxdy
= ∫from θ1 to θ2 ∫from r1 to r2 f(cosθ, sinθ) (r dr dθ)

The Attempt at a Solution

The first thing I did is sketch y = (2x-x^2)^(1/2) in xy plane.

y = (2x-x^2)^(1/2)
y^2 = 2x-x^2
0 = y^2 - 2x + x^2
0 = y^2 + (x^2-2x+1) -1
1 = y^2 + (x-1)^2 ====> circle with radius one, centered around (1,0).

I also graphed y=0, x=1, and x=2. We know that we want the region between x=1 and x=2, and it is in postive y-axis due to y = 0. so we have a quarter of a circle that we need integral. I am stuck here, I do not what the bounds for θ and r are.

I think the bounds for θ is 0 ≤ θ ≤ pi/4 but not too sure.
I have no clue how to find the r for the problem... I know it cannot be 1 ≤ r ≤ 2.

I know how to change the given function into polar but I need help finding the bounds, if someone can help me out, please, thanks in advance.

 

[cragar]

Im wondering if you should do a change of variables on that circle so you can get it centered on the origin, Because that radius will be tricky to deal with.

 

[wildleaf]

I got the bounds for this, just a min ago.
it will be 0 ≤ θ ≤ pi/4 and sec θ ≤ r ≤ 2cosθ.
thanks anways.

 

[cragar]

oh ok, you those bounds look good that's what I got too .