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

to determine the unit cost your would divide the total cost by the number sold.

7 bagels for $4.41       $4.41/7 = $.63 each at fresh side

5 bagels for $3.60         $3.60/5 = $.72 each at value mart

5x + 3y = - 4

step-by-step explanation:

the equation of a line in standard form is

ax + by = c ( a is a positive integer and b, c are integers )

express the line first of all in slope- intercept form

y = mx + c ( m is the slope and c the y-intercept )

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = (7, - 3) and (x₂, y₂ ) = (4, - 8)

m = = -

y = - x + c ← is the partial equation

to find c substitute either of the 2 points into the partial equation

using (4, - 8 ), then

- 8 = - + c ⇒ c = -

y = - x - ← in slope-intercept form

multiply all terms by 3 to eliminate the fractions

3y = - 5x - 4 ( add 5x to both sides )

5x + 3y = - 4 ← in standard form