Simple line integral problem I cant seem to get

In summary, a line integral is a type of integral used in multivariable calculus to calculate the cumulative effect of a vector field along a specific path or curve. To set up a simple line integral problem, one must define a vector field and a curve using parametric equations. There are three types of line integrals: path integrals, line integrals of the first kind, and line integrals of the second kind. The existence of a closed form solution for a line integral problem depends on the vector field being conservative. Line integrals have various applications in physics and engineering, such as calculating work done by a force and determining potential energy in electric or gravitational fields.
  • #1
gr3g1
71
0
I have to evaluate the line integral :
[itex]\oint_{}^{} (2x + y)dx + xydy[/itex] between (-1,2) and (2,5)

on the curve: [itex]y = x + 3[/itex]

So, what I did was:
[itex]\int_{-1}^{2} (3x+3)dx + \int_{2}^{5} (x^{2} + 3x)dx [/itex]

However, this is wrong and I am not sure why!
Can someone please guide me?
Thanks alot!
 
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  • #2
You are parametrizing the curve by the value of x which goes from -1 to 2. In BOTH parts. Since you've expressed the dy part of the integral in terms of x, why are you using the y limits there?
 
  • #3
Ahhh, thanks so much!
 

1. What is a line integral?

A line integral is a type of integral in multivariable calculus that calculates the cumulative effect of a vector field along a specific path or curve. It is used to calculate physical quantities such as work, circulation, and flux.

2. How do I set up a simple line integral problem?

To set up a simple line integral problem, you need to define a vector field and a curve. This can be done by using parametric equations for the curve and expressing the vector field in terms of its components. The integral is then evaluated by substituting the parametric equations into the corresponding components of the vector field and integrating over the given bounds.

3. What are the different types of line integrals?

There are three types of line integrals: path integrals, line integrals of the first kind, and line integrals of the second kind. Path integrals are defined over a curve and do not require a vector field. Line integrals of the first kind are calculated over a vector field along a curve, while line integrals of the second kind are calculated over a scalar field along a curve.

4. How do I know if my line integral problem has a closed form solution?

There is no general method for determining if a line integral problem has a closed form solution. However, if the vector field is conservative, then the line integral will have a closed form solution. This means that the integral will only depend on the endpoints of the curve and not the path taken.

5. What are some common applications of line integrals?

Line integrals have many applications in physics and engineering. Some common applications include calculating work done by a force, finding the flux of a vector field through a surface, determining the circulation of a fluid, and calculating potential energy in an electric or gravitational field.

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