Final answer:
To compute the line integral over a curve consisting of two line segments, we integrate the function over each segment separately, parameterize the path segments, substitute the parameterized coordinates into the function, compute the differentials, and then sum the integrals over all segments.
Step-by-step explanation:
To compute the line integral along the curve consisting of two line segments, we must find the integral of the function along each segment of the path. Initially, we need specific points to define our segments. Assuming we have points A and B for the first segment, and C and D for the second, we would perform the integral from A to B and then from C to D separately. We will apply the fundamental theorem for line integrals, if applicable.
As mentioned, computing a line integral over a closed path means adding the integrals over the segments that make up this path. If we are given a function to integrate, we need to parameterize each segment of the curve and then integrate the function over this parameter. Let's consider intervals such as from x₁ to x₂ and from y₁ to y₂, and apply the correct substitutions for the variables in our function.
In some cases, we need to compute the line integral of a vector field, such as ℒ E ⋅ di over a circular arc, which would involve the radius and some trigonometric functions depending on the orientation of the arc.
Step-by-Step Procedure for Computing Line Integrals
Parameterize the path segments with a suitable variable, usually t or θ.
Substitute the parameterized coordinates into the function.
Compute the differential elements dx, dy, or, if necessary, di.
Set up the integral of the function times the differential element over the interval corresponding to each segment.
Add the results of the integrals over all segments of the path.