# Solving a Line Integral Using Green's Theorem

• hbomb
In summary, the conversation revolved around finding a line integral using Green's Theorem and using the given vector field and closed curve. Some suggestions were made to correct errors in the calculations, and ultimately the correct answer was found.
hbomb
I'm having trouble on a line integral.

Assuming that the closed curve C is taken in the counterclockwise sense. Use Green's Theorem.

$$\int_C F\bullet dR$$
where F=($$x^2 + y^2$$)i + 3x$$y^2$$j
and C is the circle
$$x^2 + y^2 = 9$$

This is what I have done so far...

$$\int_0^{2\Pi} \int_0^3 \-r^2 rdrd\theta$$

$$\int_0^{2\Pi} \int_0^3 \-r^3 drd\theta$$

$$\int_0^{2\Pi} \frac{-r^4}{4} \\]_0^3d\theta$$

$$\int_0^{2\Pi} \frac{-81}{4} d\theta$$

$$\frac{-81}{4} \theta\\]_0^{2\Pi}$$

$$\frac{-81\Pi}{2}$$

The book gives the answer as $$\frac{243\Pi}{4}$$

I have no idea where I went wrong.

Theres another form of Green's theorem that might be more appropriate for this problem, try looking for it in your textbook.

Also the integral of $r^3$ is not [itex] \frac{-r^4}{4} [/tex]. No negative.

You've also managed to neglect the vector field your integrating over.

Can you show the work as to how you got

$$\int_0^{2\Pi} \int_0^3 \-r^2 rdrd\theta$$

?

You've done something strange. Applying Green theorem should yield the equality

$$\int_C F\bullet dR = \iint_D \left(\frac{\partial (x^2+y^2)}{\partial y} - \frac{\partial (3xy^2)}{\partial x} \right) dxdy$$

and that's not

$$\int_0^{2\Pi} \int_0^3 \-r^2 rdrd\theta$$

Sorry guys, I was looking at another problem and I took the derivative of the wrong force. I redid this problem and I got the correct answer. Thanks for the attempted help though.

Np.. happens to me all the time

## 1. What is Green's Theorem?

Green's Theorem is a mathematical tool used to evaluate line integrals over a closed curve in two dimensions. It relates the line integral to a double integral over the region enclosed by the curve, making it easier to solve complex line integrals.

## 2. How is Green's Theorem useful?

Green's Theorem can be used to solve line integrals that would be difficult or impossible to solve directly. It also allows for the conversion of a line integral to a double integral, which can be easier to evaluate using standard integration techniques.

## 3. What is a line integral?

A line integral is a mathematical concept that calculates the total value of a function along a given curve. It is represented by ∫C F(x,y) ds, where C is the curve, F(x,y) is the function, and ds is an infinitesimal length along the curve.

## 4. How do you solve a line integral using Green's Theorem?

To solve a line integral using Green's Theorem, first identify the curve and the function to be integrated. Then, determine the region enclosed by the curve and set up a double integral over that region. Finally, evaluate the double integral to find the value of the line integral.

## 5. What are the requirements for using Green's Theorem?

Green's Theorem can only be used to solve line integrals over closed curves in two dimensions. Additionally, the curve must be simple and smooth, meaning it does not intersect itself and has a continuous tangent vector throughout. The function being integrated must also be continuous over the region enclosed by the curve.

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