Calculus 3: Evaluate the line integral question?

Evaluate the line integral, where C is the given curve.

C (x yz)dx 2x dy xyz dz

C consists of line segments from (2, 0, 1) to (3, 3, 1) and from (3, 3, 1) to (3, 5, 3).





Can someone please walk me through this??
c (x yz) dx 2x dy xyz dz



from (2, 0, 1) to (3, 3, 1):

x = 2 t ; y = 3t ; z = 1 ; 0 t 1 ........ in parametric form

dx = dt ; dy = 3 dt ; dz = 0

c (x yz) dx 2x dy xyz dz

= ((2 t) (3t)(1)) dt 2(2 t)(3) dt (2 t)(3t)(1)(0) dt

= 2 5t 7 dt

= 5t 14t ... from 0 to 1

= 5 14

= 19



from (3, 3, 1) to (3, 5, 3):

x = 3 ; y = 3 2t ; z = 1 2t ; 0 t 1 ........ in parametric form

dx = 0 ; dy = 2 dt ; dz = 2 dt

c (x yz) dx 2x dy xyz dz

= (3 (3 2t)(1 2t))(0) dt 2(3)(2) dt (3)(3 2t)(1 2t)(2) dt

= 6 4t 8t 5 dt

= 8 t 24 t 30 t ..... from 0 to 1

= 8 24 30

= 62



c = 19 62 = 81



Answer: 81