Double integrals in polar coordinates

[compliant]

Homework Statement

Find

$$\int{\int_{D}x dA}$$

where D is the region in Q1 between the circles x²+y²=4 and x²+y²=2x using only polar coordinates.

The Attempt at a Solution

Well, the two circles give me r=2 and r=2 cos $$\theta$$, and the integrand is going to be r²cos $$\theta$$, but I have no idea how to determine the bounds of integration in this case.

 

[tiny-tim]

… I have no idea how to determine the bounds of integration in this case.

Hi compliant!

(have a theta: θ and a pi: π )

Just integrate θ from 0 to 2π (or -π and π), and integrate r between whatever values it goes between for a fixed value of θ.

 

[HallsofIvy]

I would recommend first drawing a picture. $x^2+ y^2= 4$ is, of course, a circle with center at (0,0) and radius 2. $x^2+ y^2= 2x= x^2- 2x+ y^2= 0$ or $x^2- 2x+ 1+ y^2= (x- 1)^2+ y^2= 1$ is a circle with center at (1, 0) and radius 1: it is tangent to the y-axis at (0,0) and tangent to the first circle at (2, 0). Now think in terms of polar coordinates. Both equations become very simple in polar coordinates. What drawing the graph tells you is that you will want to handle the integration in three parts: $\theta= 0$ to $\pi/2$, $\theta= \pi/2$ to $3\pi/2$, and $\theta= 3\pi/2$ to $2\pi$.


suggested, the outside radius (the upper limit of integration) is always 2 and the inner radius (the lower limit of integration, for $\theta= 0$ to $\pi/2$ is

 

[compliant]

tiny-tim, thanks for those. desperately needed.


hallsofivy, I did draw the diagram, and found that it was rather inconveniently symmetrical, which was why I got stumped. going by your suggestion, from θ = 0 to θ = π/2, I would be integrating along the right side of the curve, where the upper bound is r = 2, and the lower bound is r = 2 cos θ. I would then solve accordingly, with r² cos θ as the integrand.

I'm just wondering though, how is the left side of the curve from θ = π/2 to θ = 3π/2 and not θ = π/2 to θ = π ? And as for the third part of the curve that goes from θ = 3π/2 to θ = 2π, that's...a straight line. =/

Argh.

 

[compliant]

sorry to do this, but bump.

 

[tiny-tim]

Hi compliant !

I'm confused …

The area is between two circles, one touching both the edge and the centre of the other.

So there are two regions:

the "left" region, which is simply a semicircle, so you know the answer already, and you needn't integrate at all (though if you did, you would integrate a constant, over the whole angle π/2 to 3π/2)

and the "right" region, which is from -π/2 to π/2, which you seem to be ok with.