Double integrals over general regions.

[juronimo]

Homework Statement

The domain D is the intersection of two disks x^2 +y^2 = 1 and x^2 + (y-1)^2 =1

use polar coordinates to find the double integral ∫∫(x)dA

Homework Equations

x = rcosθ y = rsinθ r^2 = x^2 + y^2

The Attempt at a Solution

I have drawn the circles. also i set the two circle equations to equal each other.

x^2 +y^2 = x^2 + (y-1)^2

which yielded points of intersection.. [-√(3/4), (1/2)] and [√(3/4),(1/2)]

however i am unsure how to find my intervals of θ and I'm not sure of my intervals of r. I think it may be 0≤r≤1.


any help would be appreciated, even just to put me on the right track.. I have been looking over this problem for 45 minutes now

 

[juronimo]

im looking at it now. could i use simple geometry to find the intervals of theta?

Pythagorean theorem to get the angle on each side with respect to the x axis?

 

[Deveno]

it should be obvious that from symmetry, we can just let θ run from 0 to π/2, and double the result.

also, after θ = π/6, r is just a constant function r(θ) = 1.

so the real challenge is capturing that thin sliver under the line y = (1/√3)x. the lower limit of r will be 0, of course, but the upper limit will be the circle centered at (0,1).

use x = rcos(θ) and y = rsin(θ) to derive a formula for r in terms of θ for the upper circle (it should be a fairly nice function of θ), which will give you your upper limit for r.

then just integrate the polar form (times r dr dθ, as the polar form of dA) of your function from θ = 0 to θ = π/6, and r = 0 to (upper limit from above). after that, you just have a pie slice (nifty pun, eh?) of π/3 radians, to integrate your function over (and then double everything).

 

[HallsofIvy]

I agree with Deveno. Do this integral in three parts. For $\theta$ from 0 to $\pi/6$, your boundary is the upper circle, $x^2+ (y-1)^2= 1$ or $x^2+ y^2- 2y+ 1= r^2- 2r sin(\theta)+ 1= 1$ which reduces to $r^2= 2rsin(\theta)$. Since r is not 0 anywhere except (0, 0), we must have $r= 2sin(\theta)$. That is, for $\theta$ from 0 to $\pi/2$, integrate with r from 0 to $2sin(\theta)$.

For $\theta$ from 0 to $\pi- \pi/6= 5\pi/6$, the boundary is the circle $x^2+ y^2= r^2= 1$ so the "r" integral is from 0 to 1. Finally, for $\theta$ from $5\pi/6$ to $\pi$, use r from 0 to $2sin(\theta)$ again.

 

[juronimo]

I'm adding all 3 integrals up to compute my final area?

 

[juronimo]

i see it more of as, computing 0 to pi/2 and having my upperbound be my bottom circle.
and then computing 0 to pi/6 and having my upperbound be my top circle.

subtracting the second from the first and then doubling it.

 

[SammyS]

i see it more of as, computing 0 to pi/2 and having my upper-bound be my bottom circle.
and then computing 0 to pi/6 and having my upper-bound be my top circle.

subtracting the second from the first and then doubling it.

What do you mean by the "upper circle"? ... lower circle?

 

[Deveno]

by "upper circle" i believe he means the one centered at (0,1).

however, if he wants to compute an integral over an area to be subtracted from the integral over the (lower) semi-circle, the "lower bound" for r, needs to be
r = 2 sin(θ), because the area to be subtracted is the part "outside" the upper circle.

see here (and ignore the funky scaling these are circles, not ellipses): http://www.wolframalpha.com/input/?i=plot{(x^2+y^2=1),(x^2+(y-1)^2=1),(y=(1/√3)x)}

 

[juronimo]

correct, i want to compute the area of the oval type shape in the middle.

i took the integral with 0 ≤ r ≤ 2sin(θ) 0≤ θ ≤ ∏/2

∫∫ (r²cosθ) dr dθ

and got 2/3


then i took the integral
(5/6)∏ ≤θ ≤ ∏ 0 ≤ r ≤ 2sinθ

∫∫(r²cosθ) dr dθ

and got -1/24



i took the third integral.
0 ≤ θ ≤ (5/6)∏ 0≤r≤1
∫∫ (r²cosθ) dr dθ

and got -1/6.



im stumped after that, i feel as though i would add them all up.

doing so i got my total area to be 11/24

 

[Deveno]

i think you are taking the integral over the wrong regions.

above the line y = x/√3, the region in question is entirely within the unit circle. this line corresponds to θ= π/6 in the upper two quadrants.

so we first split our integral:

$$\int \int_D x dA = \int \int_D r^2cos(\theta) dr d\theta$$

$$=2\left(\int_0^{\pi/6}\int_0^{2sin(\theta)} r^2cos(\theta) drd\theta + \int_{\pi/6}^{\pi/2}\int_0^1 r^2cos(\theta) drd\theta\right)$$

that is, we are splitting half of D into:

$$D_1 = \{(x,y) : x^2+y^2 \leq 1,\ y > \frac{x}{\sqrt{3}},\ x \geq 0\}$$ and

$$D_2 = \{(x,y) : x^2 + (y-1)^2 \leq 1,\ y \leq \frac{x}{\sqrt{3}}\}$$

these are disjoint, so the integral over half of D is the integral over D₁ + the integral over D₂.

i don't believe your answer is correct.