The question I am attempting is;

Verify Greens Theorem for ∫(4x-3y²)dx+(y-4x²)dy

C

Where c is the boundary of the region defined by y = x^4 and y = x,

My problem is that I am getting two different answers.

1

I have put in y= x4 to get ∫ (4x-3x^8+x^4-4x^2)dx for this I get an answer of 11/30

X=0

0

I then put in y=x to get ∫ (5x-7x^2)dx for this I get a value of -1/6

X=1

When combined these two values equal 1/5.

Then according to Green’s thm this should equal

1 y=x

∫ ∫ {d(y-4x²)/dx – d(4x-3y²)/dy}dydx

X=0 y=x4

= ∫∫ (6y-8x)dydx = ∫ [3y^2-8xy] between y=x and y=x4

1

I get this equal to ∫(3x^2-8x^2-3x^8+8x^5)dx

X=0

Which I get as [(-5x^3)/3-(3x^9)/9+(8x^6)/6] which equals -5/3-3/9+8/6 = -2/3 ≠ 11/30

if you can spot where I’ve gone wrong or tell me what answer I should be looking for is it would be much appreciated thanks.