1. ## area of annulus

find area between 2 circles: x^2+y^2=1 and x^2+y^2=4

using (x+y) da

I got the answer of (3*pi)/2 is this a correct answer? To check I just took the difference between the two circles and then got the half of that. Does this seem right?

2. Originally Posted by caliguy
find area between 2 circles: x^2+y^2=1 and x^2+y^2=4

using (x+y) da

I got the answer of (3*pi)/2 is this a correct answer? To check I just took the difference between the two circles and then got the half of that. Does this seem right?
$\displaystyle \int\int_A(x+y)\,dA$

Use polar coordinates.

$\displaystyle x=r\cos\theta$
$\displaystyle y=r\sin\theta$
$\displaystyle dA=r\,dr\,d\theta$
$\displaystyle r=1..2$
$\displaystyle \theta=0..2\pi$

$\displaystyle \int_0^{2\pi}\int_1^2 r^2(\cos\theta+\sin\theta)\,dr\,d\theta$

When I integrate this, I got zero. It's possible I made a sign goof somewhere though.

3. The integral over the region is zero because

$\displaystyle \int\int_{\omega} x ~dA$ = ( x-center of gravity ) ( area of the region )

$\displaystyle \int\int_{\omega} y ~dA$ = ( y-center of gravity ) ( area of the region )

the center of gravity is $\displaystyle (0,0)$