# 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).

1. 1. ,

   2. ,

   3. ,

   where is the line segment from 1 to .

2. 1. ,

   2. ,

   where is the unit circle .

3. 1. ,

   2. ,

   where is the upper half of the circle with centre 0 and radius 2 traversed from 2 to .

### Answer

1. The standard parametrisation of , the line segment from 1 to , is

   hence

   1. Here , and

   2. Here , and

      (Note that this integral is different from , which from part (a)(i) is 0.)

   3. Here , and

      (Again, note that this is different from .)

2. We set out this solution in a similar style to Example 4.

   The standard parametrisation of , the unit circle , is

   hence

   1. Here , and

   2. Here , and

3. The standard parametrisation of , the upper half of the circle with centre 0 and radius 2, traversed from 2 to , is

   hence

   1. Here , and

   2. Here , and

## Exercise 9

Evaluate

for each of the following contours from 0 to .

### Answer

1. , where is the line segment from 0 to and is the line segment from to .

   We choose to use the standard parametrisations

   Then , . Hence

2. , 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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