Understanding Line Integrals and Vector Fields

To find the maximum value of the vector field

To find the length of the curve

To calculate the work done by the vector field on a particle

To determine the area enclosed by the curve

5

3

4

2

x = r sec t, y = r csc t

x = r tan t, y = r cot t

x = r cos t, y = r sin t

x = r sin t, y = r cos t

4 sin t

4 cos t

3 cos t

3 sin t

Using x = 4 sin t and y = 4 cos t

Using x = 3 sin t and y = 3 cos t

Using x = 4 cos t and y = 4 sin t

Using x = 3 cos t and y = 3 sin t

4 sin t

-4 sin t

4 cos t

-4 cos t

48 + 16 cos t sin t

32 + 16 cos t sin t

16 + 48 cos t sin t

16 + 32 cos t sin t

Integration by parts

Partial fraction decomposition

U-substitution

Trigonometric substitution

16 pi

32 pi

48 pi

64 pi

The maximum value of the vector field

The work done by the vector field on the particle

The area enclosed by the curve

The length of the curve