Question

In this and the following problem you will consider the integral

∮8ycos(2x)dx+5xydy

on the closed curve C consisting of the line segments from (0,0) to (5,7) to (0,7) to (0,0). Here, you evaluate the line integral along each of these segments separately (as you would have before having attained a penetrating and insightful knowledge of Green's Theorem), and in the following problem you will apply Green's Theorem to find the same integral. Note that you can check your answers between the two problems, because the value of the final integral will be the same (that is, the sum you find below must be equal to the final anser in the following problem).

Evaluate the integral above by finding the integral from (0,0) to (5,7), adding the integral from (5,7) to (0,7), and adding the integral from (0,7) to (0,0):

(0,0) to (5,7) = ?

(5,7) to (0,7) = ?

(0,7) to (0,0) = ?

Answer 1

The line integral over the closed curve C, the integral must be computed along each of the three line segments separately, substituting appropriate parameterizations for each segment, and then summing the results to obtain the total integral value.

Step-by-step explanation:

To evaluate the given line integral over the closed curve consisting of three line segments, we will calculate the integral separately for each segment. The closed path C is made up of three straight lines, which form a triangular path from (0,0) to (5,7) to (0,7), and then back to (0,0).

First, we start with the integral from (0,0) to (5,7). Since this is a straight line, we can parameterize it, for example, by expressing x and y as functions of a parameter t where x=5t and y=7t with t going from 0 to 1. The integral along this segment will require substituting these parameterizations into the differential components dx and dy of the integral and computing the result.

Next, for the segment from (5,7) to (0,7), y is constant at 7, and only x changes, going from 5 to 0. Therefore, dx becomes negative, and since y is constant, dy=0. This simplifies the integral along this segment because the term involving dy drops out.

Finally, for the segment from (0,7) to (0,0), x is constant at 0, and only y changes, going from 7 to 0. In this case, dx=0, so only the term involving dy needs to be considered.

By evaluating each segment separately and adding the results, we will get the value for the total line integral over the closed curve C, which should be consistent with the result from Green's Theorem for verification.