Question

Evaluate the double integral (x+y)dA, where R is the region in the first quadrant bounded by the circles x^2+y^2=4 and x^2+y^2=2y.

Answer 1

@ParthKohli @radar @Compassionate

Answer 2

start by sketching the region

Answer 3

Can we do this problem without using polar coordinates?

Answer 4

yes buy why do you want to avoid polar ? they're nice..

Answer 5

I did polar and this is what I got:
\[\int\limits_{0}^{\pi/2}\int\limits_{2}^{2\sin \theta}(r \cos \theta+r \sin \theta)r dr d \theta \]

Answer 6

that looks good, evaluate ?

Answer 7

http://www.wolframalpha.com/input/?i=%5Cint%5Climits_0%5E%7B%5Cpi%2F2%7D%5Cint%5Climits_%7B2%5Csin+%5Ctheta%7D%5E2+%28r%5E2%28+%5Ccos%5Ctheta%2B%5Csin+%5Ctheta%29%29+drd%5Ctheta

Answer 8

You're right, cartesian is much simpler :
http://www.wolframalpha.com/input/?i=%5Cint%5Climits_0%5E2%5Cint%5Climits_%7Bsqrt%282y-y%5E2%29%7D%5E%7Bsqrt%284-y%5E2%29%7D+%28x%2By%29+dxdy

Answer 9

work it `dxdy` so that finding the bounds is trivial :

```
y : 0 --> 2
x : sqrt(2y-y^2) --> sqrt(4-y^2)
```

Answer 10

Thank you!

Answer 11

yw!