Question

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

∫ c(y+e√x)dx+(2x+cosy2)dy,

C is the boundary of the region enclosed by the parabolas

y=x^2 and x=y^2

Answer 1

∫ 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