# Need help understanding the solution of this task. (vector calc)

#### uint

I'm confused about a solution in a solution manual. Given parametric curve

r(t) = a cosÂ³t + b sinÂ³t,

the task is to find the area that this curve bounds by using a line integral. The solution manual lists these 3 equations: http://a.pomf.se/nhksps.png and says that integrating any one of them from 0 to 2pi will grant the answer. Why is that?

Thanks!

#### Country Boy

Math Team
This is basically "Green's Theorem":
$$\displaystyle \oint_C Ldx+ M dy= \int_D\int \left(\frac{\partial M}{\partial x}+ \frac{\partial L}{\partial y}\right) dxdy$$

Where D is some region of the xy-plane and C is its boundary. In this case, since we are finding the area of D, which is $$\displaystyle \int_D\int 1 dxdy$$, we can take M and N to be any functions of x and y such that $$\displaystyle \frac{\partial M}{\partial x}+ \frac{\partial L}{\partial y}= 1$$.

One such would, of course, be L= 0 and M= x. That would give
$$\displaystyle \oint_C xdy= \int_D\int 1 dxdy$$

another would be L= y, M= 0. That would give
$$\displaystyle \oint_C y dx= \int_D\int 1 dxdy$$

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