Integrating $\int_{C}F\cdot dr$ with F and r Given

In summary, the conversation discusses a problem involving computing an integral, where the function is F=<e^y,xe^y,(z+1)e^z> and the path is r=<t,t^2,t^3>,0\leq t \leq 1. The individual attempts at solving the integral are described, with the conclusion that only some parts can be solved. The concept of a potential function is mentioned, and the conversation ends with the confirmation that the answer is 2e.
  • #1
Xyius
508
4

Homework Statement


The problem says to compute the following integral.

[tex]\int_{C}F\cdot dr[/tex]
Where
[tex]F=<e^y,xe^y,(z+1)e^z> \ \ and \ \ r=<t,t^2,t^3>,0\leq t \leq 1[/tex]

2. The attempt at a solution
Basically when I plug everything in, I get an integral that CANT be solved. At first I thought to use Greens theorem, but I can't because it isn't two dimensional. When I plug everything in, I get..

[tex]\int^{1}_{0} (2t^2+1)e^{t^2}+3t^5e^{t^3}+3t^2e^{t^3}dt[/tex]

The only one I can immediate see is possible to do is the last term. MAYBE the second one, but definitely not the first one. (Not one part of it anyway.)

The first part of the question asked to find to potential function and I did that, I don't know if that would be in anyway relevant to this though.

Thanks!
 
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  • #2
That's a good one! Had me scratching my head for a while. If you can get a potential function then you KNOW you can integrate it by using that. So no, you can't do e^(t^2) and you can't do t^2*e^(t^2). But you CAN do (2t^2+1)*e^(t^2). Wanna try and figure it out before I tell you??
 
  • #3
Ohh! Thats interesting! Yes let me do it now. :)
 
  • #4
Okay I got an answer of 3e. Here is my work.

http://img203.imageshack.us/img203/2750/photo1piq.jpg

Thanks a lot! :D!
 
Last edited by a moderator:
  • #5
Xyius said:
Okay I got an answer of 3e. Here is my work.

http://img203.imageshack.us/img203/2750/photo1piq.jpg

Thanks a lot! :D!

That's a little hard to read. But I get 2e.
 
Last edited by a moderator:
  • #6
Yeah sorry the resolution isn't too great. Anyway, I will post my work in detail later using Latex.
 
  • #7
Xyius said:
Yeah sorry the resolution isn't too great. Anyway, I will post my work in detail later using Latex.

You can check your answer using the potential expression, right? What does that say?
 

FAQ: Integrating $\int_{C}F\cdot dr$ with F and r Given

What is the process for integrating $\int_{C}F\cdot dr$ when F and r are given?

The process for integrating $\int_{C}F\cdot dr$ when F and r are given involves using the fundamental theorem of line integrals. This theorem states that the line integral of a vector field F along a curve C can be calculated by finding the antiderivative of F with respect to the arc length parameter and evaluating it at the endpoints of C.

How do I determine the curve C when integrating $\int_{C}F\cdot dr$ with F and r given?

The curve C can be determined by looking at the given vector field F and its corresponding position vector r. The curve C should be a path that starts at one point and ends at another point, with the position vector r representing the displacement between these two points.

Can I use any vector field F when integrating $\int_{C}F\cdot dr$ with F and r given?

Yes, you can use any vector field F as long as it is continuous and differentiable along the curve C. This means that the components of F must have continuous first-order partial derivatives with respect to the variables of integration.

What are the limits of integration when integrating $\int_{C}F\cdot dr$ with F and r given?

The limits of integration for $\int_{C}F\cdot dr$ are the starting and ending points of the curve C. These points can be determined by looking at the position vector r and finding the coordinates of the starting and ending points.

Are there any applications of integrating $\int_{C}F\cdot dr$ with F and r given in real-world scenarios?

Yes, integrating $\int_{C}F\cdot dr$ with F and r given has many applications in physics and engineering. It can be used to calculate work done by a force along a path, calculate flux through a closed surface, or calculate the circulation of a fluid. It is also commonly used in vector calculus to solve problems involving motion, electricity, and magnetism.

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