Integrate Form to Find Area of Square & Semicircle

[Mathman_]

Homework Statement

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.

Homework Equations

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=[tex]\int[/tex] 1.dxdy


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

The map
(r,[tex]\theta[/tex]) [tex]\mapsto[/tex] (x,y)
x=rcos[tex]\theta[/tex] , y=rsin[tex]\theta[/tex]

[0,2]x[0[tex]\pi[/tex]] [tex]\rightarrow[/tex] C

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

x=rcos([tex]\pi[/tex]s)
y=rsin([tex]\pi[/tex]s)

1/2Area C= [tex]\int_{\sigma}[/tex] 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. :)

 

[mighty2000]

Hello, thank you for your post. I can provide some guidance on how to approach this problem.

First, let's break down the problem into smaller parts. The window is made up of a square and a semicircle. We can find the area of each individually and then add them together to get the total area of the window.

To find the area of the square, we can use the formula A = s^2, where s is the length of one side of the square. In this case, s = 2, so the area of the square is 4 square units.

Next, we can find the area of the semicircle. We know that the formula for the area of a circle is A = πr^2, where r is the radius of the circle. Since this is a semicircle, we only need to find half of the area. The radius of the semicircle is also 2, so the area of the semicircle is π(2)^2/2 = 2π square units.

Now, to find the total area, we add the area of the square and the area of the semicircle together: 4 + 2π = 4 + 2π. This is the final answer, but we can also express it in terms of an integral using the area form w = dx ^ dy.

To do this, we need to identify the chain. In this case, the chain is the boundary of the window, which is made up of the square and the semicircle. We can represent this boundary as a 2-chain in R2, denoted by C.

Now, we can express the area of the window as the integral of the area form over the 2-chain C: ∫w = dx ^ dy over C. This integral represents the sum of the areas of the square and the semicircle.

I hope this helps to clarify the problem and provide a starting point for solving it. Let me know if you have any further questions or need more clarification. Good luck!