(906, #15) Evaluate the line integral:

where C consists of the line segment from ( 0, 0, 0 ) to ( 0, 1, 1 ), from ( 0, 1, 1 ) to ( 1, 2, 3 ), and from ( 1, 2, 3 ) to ( 1, 2, 4 ).

Solution: The path of integration is, of course, a set of three line segments starting at the origin and ending at (1,2,4):

We can break up C into three parts and we can parameterize each segment.

On C1

x= 0
y = t
z = t
t = 0..1

dx = 0
dy = dt
dz = dt

On C2

x = t
y = t + 1
z = 2 t + 1
t = 0..1

dx = dt
dy = dt
dz = 2dt

On C3

x = 1
y = 2
z = t+3
t = 0..1

dx = 0
dy = 0
dz = dt

We also break up our integral into three parts using the information above.




by C.A.H.