2.4 Further exercises
Here are some further exercises to end this section.
Exercise 8
Evaluate the following integrals (using the standard parametrisation of the path in each case).
,
,
,
where
is the line segment from 1 to
.
,
,
where
is the unit circle
.
,
,
where
is the upper half of the circle with centre 0 and radius 2 traversed from 2 to
.
Answer
The standard parametrisation of
, the line segment from 1 to
, is
hence
Here
, and
Here
, and
(Note that this integral is different from
, which from part (a)(i) is 0.)
Here
, and
(Again, note that this is different from
.)
We set out this solution in a similar style to Example 4.
The standard parametrisation of
, the unit circle
, is
hence
Here
, and
Here
, and
The standard parametrisation of
, the upper half of the circle with centre 0 and radius 2, traversed from 2 to
, is
hence
Here
, and
Here
, and
Exercise 9
Evaluate

for each of the following contours from 0 to
.

Answer
, where
is the line segment from 0 to
and
is the line segment from
to
.
We choose to use the standard parametrisations
Then
,
. Hence
, where
is the line segment from 0 to 1 and
is the line segment from 1 to
.
We choose to use the standard parametrisations
Then
,
. Hence
(Note that the integrals in parts (a) and (b) have different values.)
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