# Integration of forms

1. Dec 9, 2009

### Mathman_

1. The problem statement, all variables and given/known data

A window has the shape of a square of side 2 surmounted by a semicir-
cle. Find its area. Express the computation in terms of the integral of the area form
w = dx ^ dy over a 2-chain in R2. Identify the chain.

2. Relevant equations

3. The attempt at a solution

I don't understand how to do this...
This is what I have so far
C={(x,y) in R2| x^2 + y^2 =1}
w=dx ^ dy

Area C=$$\int$$ 1.dxdy

singular 2-cube $$\sigma$$ : [0,1]$$^{2}$$ $$\rightarrow$$ $$\textbf{R}^{2}$$ such that C=$$\sigma$$([0,1]$$^{2}$$)

The map
(r,$$\theta$$) $$\mapsto$$ (x,y)
x=rcos$$\theta$$ , y=rsin$$\theta$$

[0,2]x[0$$\pi$$] $$\rightarrow$$ C

Then $$\sigma$$ : [0,1]$$^{2}$$ $$\rightarrow$$ C:(r,s) $$\mapsto$$ (x,y)

x=rcos($$\pi$$s)
y=rsin($$\pi$$s)

1/2Area C= $$\int_{\sigma}$$ dx ^ dy

I don't know how to solve this. I've checked in many texts and online. I don't need a detailed solution. I just want to know how to compute the area of this semi circle and the 2x2 square and how to identify the chains...

Any help would be appreciated. :)

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