Calculating Line Integral of Lamda Curve

In summary, the problem involves calculating a line integral using a given curve lambda, which starts at (0,0) and follows two straight lines before ending at (2,1). The integral involves the variables x and y, and the attempt at parametrization using x=t and y=t is incorrect. Instead, using x=t+1 and y=t correctly parametrizes the line and yields the solution of 13/2.
  • #1
277
1

Homework Statement



A curve lamda starts in (0,0) and fallows a straight line from x = 0, y = 0 to x = 1, y = 0 and then another straight line to x = 2 and y = 1. Calculate the line integral I = S vdr
where S is an integral s
v = (x+3y)i + (3+y)j

Homework Equations





The Attempt at a Solution



the first line
y = 0
dy = 0
S x dx from 0 to 1
= 1/2

The second line
x = t
y = t
dx = dt
dy = dt

S (t+3t)dt + (3+t)dt
and here it stops. Both x and y are variables and x are supposed to go from 1 to 2 and y from 0 to 1.
 
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  • #2
x=t and y=t doesn't parametrize the line connecting (1,0) and (2,1), does it?
 
  • #3
then there is something I have got wronge. I have not done much of parametrization, but I was thinking a straight line from (1,0) to (2,1) had to be 45 degrees and the you used x = t and y = t which gives 45 degree angle.
 
  • #4
It's 45 degree's alright but if you put t=0, you get (0,0). How about putting x=t+1?
 
  • #5
So it is?

x = t+1
y = t
dx = dt
dy = dt
S t+1 + 3t + 3 + t dt
S 5t + 4 dt from 0 to 1
= 13/2

Thanks for all the help
 

1. What is a line integral of a Lamda curve?

The line integral of a Lamda curve is a mathematical concept that involves calculating the integral of a vector field along a curve. In simpler terms, it is a way to measure the total amount of a vector quantity (such as force or displacement) along a given path.

2. How is the line integral of a Lamda curve calculated?

The line integral of a Lamda curve is calculated by evaluating the integral of the vector field (represented by λ) along the curve. This involves breaking the curve into small segments, finding the dot product of the vector field and the tangent vector at each point, and summing them up. The result is the line integral value.

3. What is the significance of calculating the line integral of a Lamda curve?

The line integral of a Lamda curve is important in various fields of science and engineering, as it allows us to quantify and analyze the behavior of vector fields. It is especially useful in physics, where it is used to calculate quantities such as work, energy, and flux.

4. What are the different types of line integrals for Lamda curves?

There are two types of line integrals for Lamda curves: the line integral of the first kind (also known as the path integral), which is defined along a smooth curve in a two-dimensional plane, and the line integral of the second kind, which is defined in three-dimensional space and involves the use of a surface integral.

5. Are there any applications of line integrals for Lamda curves in real life?

Yes, there are many real-life applications of line integrals for Lamda curves. Some examples include calculating the work done by a force on an object moving along a curved path, determining the circulation of a fluid flow around a closed loop, and determining the mass of an object with varying density along its length.

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