# 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.)