1 Attachment(s)
Re: Area between two circles
Re: Area between two circles
Hello, nubshat!
Quote:
Find the area of the region between the circles: $\displaystyle x^2 + y^2 \,=\, 4\,\text{ and }\,x^2 + y^2 \,=\,4x$
The graph looks like this . . .
Code:

* * * * * *
*  * * *
*  *:://* *
* *:::///* *
::::/A//
* *::::////* *
**+**
* *::::1:::*2 *
::::::::
* *::::::* *
*  *::::* *
*  * * *
* * * * * *

Note that region $\displaystyle A$ is onefourth of the desired area
and $\displaystyle A$ can be found with: .$\displaystyle \int^2_1\sqrt{4x^2}\,dx$
Re: Area between two circles
Re: Area between two circles
Post #3 is correct. The integral expressions should be swapped (and 2 added) in post #2: WolframAlpha.
It is easy to find the area without calculus. Using notations from post #2, we need to find twice the area of the circular segment PLRQ. Then $\displaystyle \tan\angle POL=\sqrt{3}$, so the angle is 60 degrees. Therefore, the central angle POR of the segment is 120 degrees. The area of the corresponding circular sector is $\displaystyle (120/360)\pi r^2$, and the area of the triangle POR is $\displaystyle (1/2)PO\cdot OR\sin\angle POR =\sqrt{3}$. From here, the area of PLRQ is $\displaystyle (4/3)\pi\sqrt{3}$, and the final answer is twice that.