Find the area of the lune formed (Using calc please)

[anelephant09]

Hi all,

Homework Statement

Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R.

http://mathhelpforum.com/attachments/calculus/19696-find-area-untitled.png

Homework Equations

I know we have to use arcsin which is $$ \frac {1}{\sqrt{a^{2}-x^{2}}}$$

The Attempt at a Solution

I tried plotting the circles on a coordinate axis with the bigger circle centered at the origin and the smaller circle centered at a point (0,b) but I am not really sure where to go from here... If this was geometry, it would have been easy, but we have to use calc on this problem.

So far, i started with $$r \sqrt{R^{2}-r^{2}} +\frac{pir^{2}}{2}$$
but I am stuck now. Can anyone provide a push in the right direction for me?
Thanks in advance.

 

[SammyS]

Hi all,

Homework Statement

Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R.
http://mathhelpforum.com/attachments/calculus/19696-find-area-untitled.png

Homework Equations

I know we have to use arcsin which is $$ \frac {1}{\sqrt{a^{2}-x^{2}}}$$

The Attempt at a Solution

I tried plotting the circles on a coordinate axis with the bigger circle centered at the origin and the smaller circle centered at a point (0,b) but I am not really sure where to go from here... If this was geometry, it would have been easy, but we have to use calc on this problem.

So far, i started with $$r \sqrt{R^{2}-r^{2}} +\frac{pir^{2}}{2}$$but I am stuck now. Can anyone provide a push in the right direction for me?
Thanks in advance.

Hello anelephant09. Welcome to PF !

Your image was not visible, but I see it now.

Here it is again:

You need to know the separation distance between the centers of the circles .

Added in Edit:
Oh! I see from the diagram, you can figure out the separation.

 

[E'lir Kramer]

Area of circles by integration

Whoops, meant to make a new thread.