1. ## double integral

Use a double integral in polar coordinates to find the area of the region inside the circle x^2 + y^2 =4 and to the right of the line x=1.

Thank you very much.

2. I still don't know how to solve this question. If you can help, please do. Thank you very much.

3. Hello, kittycat!

Use a double integral in polar coordinates to find the area of the region
inside the circle $\displaystyle x^2 + y^2 \:=\:4$ and to the right of the line $\displaystyle x=1$.
Code:
                |Y
|
* * *   |
*     |     *P
*       |     |:*
*        |     |::*
|     |:::
*         |     |:::*
- - * - - - - + - - +:-:* - -
*         |O   1|:::*2  X
|     |:::
*        |     |::*
*       |     |:*
*     |     *Q
* * *   |
|

The polar equation of the circle is: .$\displaystyle r \,=\,2$
The polar equation of $\displaystyle x = 1$ is: .$\displaystyle r \,=\,\sec\theta$

Draw radii $\displaystyle OP$ and $\displaystyle OQ$.
We find that: .$\displaystyle P(1,\,\sqrt{3}),\;Q(1,\,\text{-}\sqrt{3})$
. . and that: $\displaystyle \angle POX \,=\,\angle QOX \,=\,\frac{\pi}{3}$

The double integral is: .$\displaystyle A \;=\;\int^{\frac{\pi}{3}}_{\text{-}\frac{\pi}{3}} \int^2_{\sec\theta} r\,dr\,d\theta$

4. okay , i got it now. Many thanks )

5. Hey Kittycat:

Just for fun (yes, we mathnerds think it's fun), here it is in rectangular coordinates.

$\displaystyle 2\int_{1}^{2}\int_{0}^{\sqrt{4-x^{2}}}dydx$