# Calculate Integral of F*n on Square Curve C

• yolanda
In summary, the homework statement is that you need to calculate the integral of F*n ds for F(x,y)=xi+yj across the square curve C with vertices (1,1), (-1,1), (-1,-1), and (1,-1). The attempt at a solution is to rewrite this as \intab F(r(t))*r'(t)) dt. To do this, you need to parameterize x and y.
yolanda

## Homework Statement

First off, sorry this isn't symbolized correctly. I've wrestled with this for a half hour, so here it is in crude type:

Calculate $$\int$$c F*n ds for F(x,y)=xi+yj across the square curve C with vertices (1,1), (-1,1), (-1,-1), and (1,-1).

above

## The Attempt at a Solution

I've found that this can be rewritten as $$\int$$ab F(r(t))*r'(t)) dt

I know my limits are from -1 to 1. I am having trouble parameterizing x and y. I have come up with x=t, but I don't know what y could be... y=$$^{+}_{-}$$t?

If someone wouldn't mind explaining this problem, and how they'd attack it, I sure would appreciate the help. Thanks in advance. A step by step solution would help most, but I'll take what I can get :)

Because the path is not "smooth", break it into smooth parts. That is, do the lines from
(1) (1,1) to (-1,1)
(2) (-1,1) to (-1,-1)
(3) (-1,-1) to (1,-1)
(4) (1,-1) to (1,10)
separately.

For example, the line from (1,1) to (-1,1) is simply "y= 1" with x going from 1 to -1.
You could just use x itself as parameter with x going from 1 to -1. Since y= 1, dy= 0.
If you don't like integrating from 1 down to -1, you could let x= -t so that dx= -dt and you are integrating with respect to t from -1 to 1.
Or you could take x= -2t+ 1 so that dx= -2dt and integrate with respect to t from 0 to 1. The crucial point is that at every point on this line y= 1 so dy= 0.

Do the same kind of thing for the other three lines.

Okay, that makes more sense. So, to get my final answer I should be adding the results of the 4 line integrals, right?

So far, for the line segment from (1,1) to (-1,1) I'm getting:

x=t y=1

r(t)=ti+j
r'(t)=i+0
F(r(t))=ti+j

So, $$\int$$$$^{-1}_{1}$$<ti,j>dot<i,0> = $$\int$$$$^{-1}_{1}$$t dt = t$$^{2}$$/2 |$$^{-1}_{1}$$ = 0

How am I doing here? Everything look okay?

## What is the definition of an integral?

An integral is a mathematical concept that represents the area under a curve on a given interval. It is a fundamental tool in calculus and is often used to calculate quantities such as displacement, velocity, and acceleration.

## What is the formula for calculating an integral?

The formula for calculating an integral is ∫f(x)dx, where f(x) is the function being integrated and dx represents the infinitesimal change in the independent variable, usually denoted as x. This formula is known as the indefinite integral, and it represents the antiderivative of the function f(x).

## What is the process for calculating the integral of F*n on a square curve C?

The process for calculating the integral of F*n on a square curve C involves breaking the curve into small, rectangular sections and finding the area of each section. Then, by summing the areas of all the sections, the total integral of F*n on the curve C can be determined. This process is known as the Riemann sum.

## Are there any special cases to consider when calculating the integral of F*n on a square curve C?

Yes, there are a few special cases to consider when calculating the integral of F*n on a square curve C. One case is when the curve C is not a perfect square, in which case the integral can still be calculated by approximating the area using the Riemann sum. Another case is when the function F is not continuous on the curve C, in which case the integral may not exist.

## What are some real-world applications of calculating the integral of F*n on a square curve C?

The integral of F*n on a square curve C has many real-world applications, such as calculating the work done by a force on an object, finding the total distance traveled by a moving object, and determining the total cost of a changing variable over a given time interval. It is also used in physics, engineering, and economics to solve various problems involving rates of change.

• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
14
Views
693
• Calculus and Beyond Homework Help
Replies
3
Views
641
• Calculus and Beyond Homework Help
Replies
9
Views
272
• Calculus and Beyond Homework Help
Replies
5
Views
936
• Calculus and Beyond Homework Help
Replies
2
Views
503
• Calculus and Beyond Homework Help
Replies
3
Views
743
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
415