Line Integral (where the path is not single valued)

In summary, the line integral can be solved using polar coordinates, where x = cos(theta), y = sin(theta), and theta ranges from pi/2 to 3pi/2. The resulting integral is -2 times the integral of SQRT(1 - x^2) from 0 to 1, due to the orientation of the curve.
  • #1
Neoleachster
1
0
I'm a bit confused as to how to solve this line integral:

Evaluate I = The integral of (x+y)dx from A (0,1) to B (0,-1) along the semi-circle
x^2 + y^2 = 1 for for x is equal to or greater than 0.

So far have have got:

if x^2 + y^2 = 1 then y= + SQRT of (1 - x^2)

which I believe boils down to : 2 times the integral of SQRT (1 - x^2) from 0 to 1.

I'm very confused how to finish this off and would appreciate any help.
 
Physics news on Phys.org
  • #2
Use polar coordinates! Ie: x = cos(theta), y = sin(theta), theta going from pi/2 to 3pi/2. (And include an overall negative sign due to the orientation of the curve)
 

1) What is a line integral where the path is not single valued?

A line integral where the path is not single valued is a type of integral used in multivariable calculus to calculate the total value of a scalar or vector field along a curve that is not a single continuous line. This means that the path may have more than one starting and ending point, or it may overlap with itself.

2) How is a line integral where the path is not single valued different from a regular line integral?

The main difference between a line integral where the path is not single valued and a regular line integral is that in the former, the path must be defined parametrically in order to properly calculate the integral. This means that the path is described by a set of equations rather than a single equation.

3) What are some real-world applications of line integrals where the path is not single valued?

Line integrals where the path is not single valued have many important applications in physics and engineering. They are used to calculate the work done by a force along a non-linear path, the flow of a fluid along a complex curve, and the circulation of a vector field around a closed path. They are also used in the study of electromagnetic fields and in the calculation of electric and magnetic flux.

4) How is a line integral where the path is not single valued calculated?

To calculate a line integral where the path is not single valued, the path must first be defined parametrically. Then, the integral is calculated by evaluating the function being integrated at each point along the curve and multiplying it by the derivative of the path with respect to the parameter. These values are then summed up to find the total value of the integral.

5) Are there any limitations to using line integrals where the path is not single valued?

One limitation of using line integrals where the path is not single valued is that the path must be smooth and continuously differentiable. This means that it cannot have any sharp corners or cusps. Additionally, the parametric equations used to define the path must be well-behaved and not have any singularities, as this can lead to incorrect or undefined results.

Similar threads

I Is My Calculus of Variations Approach Correct?
    • Calculus
Replies
12
Views
1K
B Question about the limits of integration in a change of variables
    • Calculus
Replies
2
Views
290
B Integration by parts of inverse sine, a solved exercise, some doubts...
    • Calculus
Replies
4
Views
910
B Why is this definite integral a single number?
    • Calculus
Replies
20
Views
2K
B Question about change of variables
    • Calculus
Replies
29
Views
717
B Having trouble deriving the volume of an elliptical ring toroid
    • Calculus
Replies
4
Views
348
B Calculating shaded area: I'm getting discrepancies between methods
    • Calculus
Replies
3
Views
329
I Substitution in a definite integral
    • Calculus
Replies
6
Views
1K
A Non solvable integral? (dx/dt)^2 dt
    • Calculus
Replies
3
Views
2K
B Trigonometric substitution, a case I'd like to share
    • Calculus
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top