Answer:
7/3
Step-by-step explanation:
2x - 3y = 6
when x =-1/2 we have
2(-1/2) - 3y = 6
3y = 6 + 2(1/2) = 7
y = 7/3 answer
By Green's theorem, the integral of
along
is
![\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(2x)}{\partial x}-\frac{\partial(3x-4y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy=6\iint_D\mathrm dx\,\mathrm dy](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint_C%5Cvec%20F%5Ccdot%5Cmathrm%20d%5Cvec%20r%3D%5Ciint_D%5Cleft%28%5Cfrac%7B%5Cpartial%282x%29%7D%7B%5Cpartial%20x%7D-%5Cfrac%7B%5Cpartial%283x-4y%29%7D%7B%5Cpartial%20y%7D%5Cright%29%5C%2C%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy%3D6%5Ciint_D%5Cmathrm%20dx%5C%2C%5Cmathrm%20dy)
which is 6 times the area of
, the region with
as its boundary.
We can compute the integral by converting to polar coordinates, or simply recalling the formula for a circular sector from geometry: Given a sector with central angle
and radius
, the area
of the sector is proportional to the circle's overall area according to
![\dfrac A{\frac\pi3\,\rm rad}=\dfrac{16\pi}{2\pi\,\rm rad}\implies A=\dfrac{8\pi}3](https://tex.z-dn.net/?f=%5Cdfrac%20A%7B%5Cfrac%5Cpi3%5C%2C%5Crm%20rad%7D%3D%5Cdfrac%7B16%5Cpi%7D%7B2%5Cpi%5C%2C%5Crm%20rad%7D%5Cimplies%20A%3D%5Cdfrac%7B8%5Cpi%7D3)
so that the value of the integral is
![\dfrac{6\times8\pi}3=\boxed{16\pi}](https://tex.z-dn.net/?f=%5Cdfrac%7B6%5Ctimes8%5Cpi%7D3%3D%5Cboxed%7B16%5Cpi%7D)
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