# Line Integral does not match Greens Theorem?

• PhantomPower
In summary, the conversation discusses the evaluation of a line integral on a square curve in the anti-clockwise sense. The attempt at a solution involves breaking the integral into four parts and applying Green's theorem, but a mistake is made in the calculation due to a negative sign. The mistake is pointed out and corrected by another user.
PhantomPower

## Homework Statement

To evaluate the following line integral where the curve C is given by the boundary of the square 0 < x < 2 and 0 < y < 2 (In the anti clockwise sense):

$\oint (x+y)^2 dx + (x-y)^2 dy$

## The Attempt at a Solution

Firstly it is noted that for a square ABDE :
Between AB, dy = 0 ,y = 0
Between BD, dx = 0 ,x = 2
Between DE, dy = 0 ,y = 2
Between EA, dx = 0 ,x =0

Thus : $\int^2_0 x^2 dx + \int^2_0 (2-y)^2 dy + \int^0_2 (x+2)^2 dx + \int^0_2 -y^2 dy$
Giving $\frac{x^3}{3} |^2_0 + \frac{-(2-y)^3}{3} |^2_0 + \frac{(x+2)^3}{3} |^0_2 + \frac{-y^3}{3} |^0_2$

Evaluating gives 10.6? but applying greens theorem gives -16. Can anyone spot my mistake - Probaly a negative sign?

Thanks very much.
P.s sorry for typos this keyboard is broken

Hi PhantomPower!

erm

∫ (2 - y)2 is the same as ∫ (y - 2)2 !

(similary for (-y)2)

Ooops.

Thanks very much - been a long day...

## 1. What is a line integral?

A line integral is a mathematical concept in multivariable calculus that involves finding the integral of a function along a curve or a path in a two or three-dimensional space.

## 2. What is Greens Theorem?

Greens Theorem is a mathematical theorem that relates the line integral of a vector field over a closed curve to the double integral of the curl of the vector field over the region enclosed by the curve.

## 3. Why might the line integral not match Greens Theorem?

The line integral may not match Greens Theorem if the curve is not a closed loop, the vector field is not continuous, or if the region enclosed by the curve has holes or self-intersections.

## 4. How can I check if my line integral matches Greens Theorem?

To check if your line integral matches Greens Theorem, you can evaluate both sides of the theorem separately and compare the results. If they are equal, then the line integral matches Greens Theorem.

## 5. What are some practical applications of line integrals and Greens Theorem?

Line integrals and Greens Theorem have various applications in physics, engineering, and other fields. They are used to calculate work done by a force, fluid flow, and electric fields, among others.

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