Question

Evaluate the line integral, where c is the given curve. c (x + yz) dx + 2x dy + xyz dz, c consists of line segments (1, 0, 1) to (2, 4, 1) and from (2, 4, 1) to (2, 6, 3)

Answer 1

Parametrization of the first segment:
x=t+1, y=4t, z=1 wherein t is on the segment [0,1].
Second segment:
x=2, y=2t+4, z=2t+1 and again t is in [0,1].
Compute the derivatives like this:
First segment: dx=1dt, dy=4dt, dz=0
Second segment:dx=0, dy=2dt, dz=2dt.
Using the above variables, the given integral becomes like this:
$\int_0^1(((t+1)+4t)dt+2(t+1)4dt)+(4\times2dt+2(2t+4)(2t+1)2dt)$
The above integral is classical, and simple computation we obtain:
 $\frac{389}{6}$