Green's theorem

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5b. Green's theorem gives us a possibility to compute the area of a plane region integrating along its boundary. Actually, it can help for more complex tasks then computing area. There are two (and more) forms of that integral,

`|oint_C x dy|` and `|oint_C y dx|,` where `C` is the bounding curve.

In our case `y` better suits as the independent variable, so compute `oint_C x dy.` The curve consists of two parts, and the integral is the sum of two integrals, `int_(C_1)` and `int_(C_2),` where `C_1` is the segment and `C_2` is the semi-circumference. Note that to get round the boundary in the right direction, we have to integrate over `C_1` from the larger `y` to the smaller.

The rest is simple,

`int_(C_1) = int_1^(-1) x dy =int_1^(-1) 1 dy = -2,`  and  `int_(C_2) =int_(-1)^(1) x dy =int_(-1)^(1) y^2 dy = 2/3.`

So the area is `|-2+2/3|` = 4/3.

 

The question 5a should be asked separately. If you would ask it again, please make clear what symbol is after `(x+2y^2).`  `(-j)` which means unit vector of the y-axis?

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