# Math Help - Green's theorem

1. ## Green's theorem

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.

2. Originally Posted by jarvis5
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
You have go integrate along the curve [tex]y= x^4, not just replace y by $x^4$. If [tex]y= x4, then $dy= 4x^2 dx$, not just "dx"
[/quote] 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.[/QUOTE]