# Line Integral Help: Evaluating F ds on a Curve in 1st Quadrant

• Kuma
In summary, the conversation discusses trying to evaluate a line integral with a given vector field and path, using Green's Theorem to find a solution of 0 due to the vector field being conservative. The concept of a conservative vector field is also briefly mentioned.
Kuma

## Homework Statement

Trying to evaluate the following line integral:

integral F ds where F = (6(x^2)(y^2), 4(x^3)(y) + 5y^4)
and the path is the boundary curve of the first quadrant below y = 1-x^2 in a clockwise direction.

## The Attempt at a Solution

So since the curve is piecewise smooth closed simple and closed curve I can use greens theorem. Simply put I get an answer as 0 since dF1/dy = dF2/dx. Is that right?

Kuma said:

## Homework Statement

Trying to evaluate the following line integral:

integral F ds where F = (6(x^2)(y^2), 4(x^3)(y) + 5y^4)
and the path is the boundary curve of the first quadrant below y = 1-x^2 in a clockwise direction.

## The Attempt at a Solution

So since the curve is piecewise smooth closed simple and closed curve I can use greens theorem. Simply put I get an answer as 0 since dF1/dy = dF2/dx. Is that right?

I would agree with that.

Your vector field is conservative. So the line integral would be zero over any closed curve (even non-simple closed curve!). A vector field is conservative if it is the gradient of a potential function.

## 1. What is a line integral?

A line integral is a type of integral that is calculated along a curve in a specific direction. It involves finding the area under a curve or the work done along a path.

## 2. How do you evaluate a line integral?

To evaluate a line integral, you first need to parameterize the curve into a set of equations. Then, you can use the appropriate formula to calculate the integral, which may involve finding the limits of integration and solving the integral using techniques such as substitution or integration by parts.

## 3. What is the significance of evaluating a line integral in the 1st quadrant?

Evaluating a line integral in the 1st quadrant can help to find the work done along a path or the area under a curve in a specific region. It is also useful in solving problems related to physics, engineering, and other scientific fields that involve working with vector fields.

## 4. What are some common applications of line integrals?

Line integrals have various applications in physics, engineering, and other scientific fields. They are commonly used to calculate the work done by a force along a path, the electric field around a charged object, and the magnetic field around a current-carrying wire. They are also used in fluid mechanics to calculate the flow of a fluid along a curve.

## 5. How does the direction of the curve affect the line integral?

The direction of the curve is an important factor in evaluating a line integral. The direction determines the orientation of the curve and affects the limits of integration and the sign of the integral. It is important to specify the direction when calculating a line integral to ensure an accurate result.

Replies
4
Views
1K
Replies
12
Views
1K
Replies
3
Views
1K
Replies
10
Views
1K
Replies
12
Views
3K
Replies
2
Views
611
Replies
6
Views
1K
Replies
6
Views
2K
Replies
2
Views
779
Replies
3
Views
1K