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

∫∫∫w f(x,y) dxdy
= ∫from θ1 to θ2 ∫from r1 to r2 f(cosθ, sinθ, z) (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 positive 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 adavance.

 

[vela]

You might try first using the substitution x' = x-1 to shift the center of the circle to the origin. Then it's pretty straightforward to change to polar coordinates.

 

[Harrisonized]

Hmm. This seems to be a repeat thread. Nvrm.