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).
a.
i.,
ii.,
iii.,
where is the line segment from 1 to .
b.
i.,
ii.,
where is the unit circle .
c.
i.,
ii.,
where is the upper half of the circle with centre 0 and radius 2 traversed from 2 to .
Answer
a.The standard parametrisation of , the line segment from 1 to , is
hence
i.Here , and
ii.Here , and
(Note that this integral is different from , which from part (a)(i) is 0.)
iii.Here , and
(Again, note that this is different from .)
b.We set out this solution in a similar style to Example 4.
The standard parametrisation of , the unit circle , is
hence
i.Here , and
ii.Here , and
c.The standard parametrisation of , the upper half of the circle with centre 0 and radius 2, traversed from 2 to , is
hence
i.Here , and
ii.Here , and
Exercise 9
Answer
a., where is the line segment from 0 to and is the line segment from to .
We choose to use the standard parametrisations
Then , . Hence
b., 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.)