Measuring the Length of a Parabolic Path with Line Integral

In summary, a line integral is a more complex concept than finding the area under a curve in 2 dimensions. It involves integrating the differential of arc length, which is given by √(1+(dy/dx)2)dx, between two values of x. This is an application of integration in 2 dimensions, while line integrals are done in 3 dimensions or more.
  • #1
wasi-uz-zaman
89
1
Hi experts
what is line integral - for example if I can draw graph of parabola and i can calculate the area under the graph. But how can i measure the length of parabolic path.
 
Physics news on Phys.org
  • #2
Let s be arc length. Then the differential of arc length is given by:

ds=√(dx2+dy2)=√(1+(dy/dx)2)dx.

To get the arc length between 2 values of x, integrate ds between those values.
 
  • #3
Two entirely different concepts
A line integral is more complex idea than the area under a curve in 2 dimensions.
It is done in 3 dimensions [or more]
The arc length is an application of integration in 2 D and the formula was given to yoou in the previous post.
 
  • #4
thanks a lot you have solved my query.
 

1. What is a parabolic path?

A parabolic path is a curved path that follows the shape of a parabola, which is a mathematical curve with a U-shape. It is formed by the motion of an object that is under the influence of gravity, where the vertical acceleration is constant and the horizontal acceleration is zero.

2. Why is line integral used to measure the length of a parabolic path?

Line integral is used to measure the length of a parabolic path because it takes into account the change in direction of the path. Since a parabolic path is curved, the distance traveled along the curve is different from the straight-line distance. Line integral considers this change in direction and calculates the actual distance traveled along the curve.

3. How is line integral calculated for a parabolic path?

The line integral for a parabolic path is calculated by integrating the arc length formula along the path. This involves dividing the path into small segments, calculating the length of each segment, and then summing up all the lengths to get the total length of the path. The formula for arc length involves the derivative of the curve, which is used to calculate the slope of the curve at each point.

4. Can line integral be used to measure the length of any curve?

Yes, line integral can be used to measure the length of any curve. It is a general method for calculating the length of any curve, regardless of its shape or complexity. However, for more complex curves, the calculation process may be more difficult and require advanced mathematical techniques.

5. Are there any limitations to using line integral for measuring the length of a parabolic path?

One limitation of using line integral for measuring the length of a parabolic path is that it assumes the path is smooth and continuous. If the path has sharp turns or discontinuities, the calculation may not be accurate. Additionally, line integral may not be suitable for calculating the length of a parabolic path in three-dimensional space, as it only considers the two dimensions of the curve.

Similar threads

  • Calculus
Replies
5
Views
1K
Replies
8
Views
265
Replies
10
Views
3K
Replies
1
Views
3K
Replies
14
Views
2K
Replies
8
Views
147
Replies
24
Views
2K
Replies
8
Views
1K
  • Calculus
Replies
14
Views
1K
Back
Top