(906, #11) Evaluate the line integral:

(906, #11) Evaluate the line integral

where C is the line segment from (1,0,1) to (0,3,6).

Solution: The vector form of the line C is <x, y, z> = <1,0,1> + <-1, 3,5> t, Thus the parametric equations are

x = 1 - t
y = 3 t
z = 1 +5 t
t = 0..1.

In addition, on C,

ds = sqrt(1² + 3² + 5²) = sqrt(35)
x y² z = (1-t)(9 t²)(1 + 5 t) = 9 t² (1 + 4 t - 5 t²)

Consequently,

by C.A.H.