Consider the line integral Z C (sin x dx + cos y dy), where C consists of the top part of the circle x 2 + y 2 = 1 from (1, 0) t o (−1, 0), followed by the line segment from (−1, 0) to (2, −π). Evaluate this line integral in two ways:
1 answer:
Direct computation :
Parameterize the top part of the circle by
with , and the line segment by
with . Then
Using the fundamental theorem of calculus :
The integral can be written as
If there happens to be a scalar function such that , then is conservative and the integral is path-independent, so we only need to worry about the value of at the path's endpoints.
This requires
So we have
which means is indeed conservative. By the fundamental theorem, we have
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