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

a4 - 3a2 - 1

step-by-step explanation:

step   1   :

equation at the end of step   1   :

  ((a4) -   3a2) -   1

step   2   :

trying to factor by splitting the middle term

2.1     factoring   a4-3a2-1  

the first term is,   a4   its coefficient is   1 .

the middle term is,   -3a2   its coefficient is   -3 .

the last term, "the constant", is   -1  

step-by-step explanation :

we know that

the measurement of the outer angle is the semi-difference of the arcs which comprises

in this problem

m< xzy is the outer angle

substitute the values

the measure of angle xoy is equal to

> by central angle

substitute the value