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

Oct 2014
42
1
Norway
I'm confused about a solution in a solution manual. Given parametric curve

r(t) = a cos³t + b sin³t,

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
Jan 2015
3,261
899
Alabama
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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