A survey asked 200 students to name their favorite fruit. The table shows the results of the survey. How many more students chos
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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