Help understanding line integral solution?

In summary, a line integral is a type of integral used in multivariate calculus to calculate the cumulative effect of a vector function along a specific path. It differs from a regular integral as it takes into account the direction of the path. The direction of the path is significant as it determines the sign of the integral. A line integral is calculated by breaking the path into segments and integrating the vector function along each segment. Some real-world applications of line integrals include calculating work, electric or magnetic flux, fluid flow, and probabilities in statistics.
  • #1
Chandasouk
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http://img2.imageshack.us/img2/5061/14983795.jpg [Broken]

I have no idea how they simplified the integral to the second step.
 
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  • #2
That looks like a line integral along a parameterized curve c(t) = <x(t),y(t),z(t)>.

You didn't copy the part that gives the formula for c(t). And remember that

ds = ||c'(t)|| dt
 

1. What is a line integral?

A line integral is a type of integral used in multivariate calculus to calculate the cumulative effect of a vector function along a curve or path. It represents the area under a curve in a multi-dimensional space.

2. How is a line integral different from a regular integral?

A line integral is different from a regular integral in that it calculates the cumulative effect of a vector function along a specific path, rather than over a two-dimensional region. It also takes into account the direction of the path, whereas a regular integral does not.

3. What is the significance of the direction of the path in a line integral?

The direction of the path in a line integral is important because it determines the sign of the integral. If the path is traversed in the direction of the vector function, the integral will have a positive value. If the path is traversed in the opposite direction, the integral will have a negative value.

4. How is a line integral calculated?

A line integral is calculated by parameterizing the curve or path and then integrating the vector function along the parameterized path. This involves breaking the curve into small segments and calculating the effect of the vector function on each segment, then summing these individual effects to get the total integral.

5. What are some real-world applications of line integrals?

Line integrals have many applications in physics, engineering, and other fields. They can be used to calculate work done by a force, calculate electric or magnetic flux, and determine the amount of fluid flow through a given path. They are also used in computer graphics to render 3D images and in statistics to calculate probabilities of multi-dimensional events.

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