# Last line integral problem (hopefully)

• PhysicsMajor
In summary, the conversation discusses the concept of independence of path in line integrals and the use of exact differentials. It is stated that a line integral is independent of path if there exists a function U that satisfies the equation dU = Fdr, and this is further explained through the chain rule and the cross derivative test. The conversation also mentions the possibility of the subject being known by a different name.
PhysicsMajor
Greetings again,

Show that for F(x,y)=<2xy-3, x^(2)+4y^(3)+5> the line integral F(x,y).dr is independant of path. Then evaluate the line integral for any curve C with initial point (-1,2) and the terminal point (2,3).

Thanks again, you all have been very helpful.

has your class covered exact differentials?

i don't believe so unless its known by another name?

I'll post a detailed explanation tomorrow (if no one else has by then). I doubt you would know the subject by a different name.

Last edited:
A line integral is independent of path if there exists a function U, that Fdr is it's exact (total) differential. In this case U=x^2y-3x+5y+y^4.
dU=(2xy-3)dx+(x^2+4y^3+5)dy=Fdr.

An "exact differential" fdx+ gdy (a physics major may prefer to think of it as a "conservative force field") is, as Oggy said, one such that there exist a function U such that dU= fdx+ gdy. By the "chain rule", $$dU= \frac{\partial U}{\partial x}dx+ \frac{\partial U}{/partial y}$$ so we must have $$f= \frac{\partial U}{/partial x}$$ and [tex]g= \frac{\partial U}{/partial g}. A quick way to test that is to use the "cross derivative test: If the second partials are continuous, that requires that
[tex]\frac{\partial f}{/partial y}= U_{xy}= U_{yx}= \frac{\partial g}{\partial x}[/itex].

## 1. What is a last line integral problem?

A last line integral problem is a type of mathematical problem that involves finding the value of an integral along a specific curve or path. The curve or path is referred to as the "last line" because it is the final segment in the integration process.

## 2. How is a last line integral problem different from a regular integral?

In a regular integral, the path of integration is not specified and can be any curve or surface. However, in a last line integral problem, the path is predetermined and must be followed to find the solution.

## 3. What are some applications of last line integral problems?

Last line integral problems are commonly used in physics and engineering to calculate physical quantities such as work, electric potential, and fluid flow. They are also used in economics and finance to model and analyze economic systems.

## 4. How do I solve a last line integral problem?

To solve a last line integral problem, you must first parameterize the given curve or path. Then, you can use the fundamental theorem of calculus to evaluate the integral along the path. Finally, you can substitute the values of the parameters to obtain the final solution.

## 5. What are some tips for tackling a last line integral problem?

Some tips for solving last line integral problems include carefully identifying the path of integration, using appropriate parameterizations, and checking for symmetry or other special properties that may simplify the problem. It is also important to review the fundamental theorem of calculus and practice with various examples to become more familiar with the process.

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