Question

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

Answer 1

Answer:

∫ C ( y + e√x) dx  +  ( 2x + cosy² ) dy = 1/3

Step-by-step explanation: See Annex

Green Theorem establishes:

∫C ( Mdx  +  Ndy )  = ∫∫R ( δN/dx  -  δM/dy ) dA

Then

∫ C ( y + e√x) dx  +  ( 2x + cosy² ) dy

Here

M = 2x  + cosy²           δM/dy  =  1

N = y + e√x                 δN/dx  =  2

δN/dx  -  δM/dy  =  2  -  1   = 1

∫∫(R) dxdy   ∫∫ dxdy

Now integration limits  ( see Annex)

dy  is from   x  = y²    then     y = √x    to  y = x²   and for dx

dx   is from 0   to  1 then

∫ dy    = y | √x   ;   x²      ∫dy    =  x² - √x

And

∫₀¹ ( x² - √x ) dx    =  x³/3  - 2/3 √x |₀¹    =   1/3 - 0

∫ C ( y + e√x) dx  +  ( 2x + cosy² ) dy = 1/3