Use Green's theorem to compute the area inside the ellipse x232+y2142=1x232+y2142=1. Use the fact that the area can be written as ∬Ddxdy=12∫∂D−y dx+x dy . ∬Ddxdy=12∫∂D−y dx+x dy . Hint: x(t)=3cos(t)x(t)=3cos⁡(t).

Answer / Explanation

First, let us define what the green theorem is:

This is a mathematical term used to give the closed gaps between the line integral of a two-dimensional vector fields over a closed path in the plane and the double integral over the region it encloses.

If we now refer back to the question asked to used the Green theorem to compute the area inside the ellipse,

We first recall the Green theorem equation which states thus:

By Green’s Theorem,

            ∫[] (Pdx+Qdy) = ∬[] (∂Q/∂x−∂P/∂y) dA  

Where p =

P = −y,

Q = x

Then,

         gives ∫ [] (−ydx + xdy) = ∬[] (1+1) dA = 2(area of )   …   (i)

Parameterize ellipse as

               x=2cost, y=13sint, 0≤t≤2π so dx=−2sint.dt, dy=13cost.dt

Then from (i), area = ½ ∫ (26sin²t+26cos²t)dt, [t=0,2π]

                               = 13 ∫ dt, [t=0,2π] = 26π

         Where,

                              x = 8cos³t, y=8sin³t, 0 ≤ t ≤ 2π

By Green’s Theorem, area

                          = ½ ∫[] (−ydx+xdy) where is enclosing contour,

                            acw dx = −24cos²tsint.dt, dy = 24sin²tcost.dt

area = ½ ∫ (192cos²tsin⁴t+192sin²tcos⁴t)dt, [t=0,2π]  

                    =  96 ∫ cos²tsin²tdt, [t=0,2π] = 24 ∫ sin²(2t)dt, [t=0,2π]  

                               = 12 ∫ (1−cos(4t))dt, [t=0,2π]

                                                  = 24π

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