Final answer:
To evaluate the line integral ∫∫∫ zdx + xzdy + xydz over the line segment from (1,1,1) to (3,2,0), we need to parameterize the line segment. After substituting the parameterization into the integral and evaluating, the line integral is equal to 1/2.
Step-by-step explanation:
To evaluate the line integral ∫∫∫ zdx + xzdy + xydz over the line segment from (1,1,1) to (3,2,0), we need to parameterize the line segment. Let's consider the parameterization:
x = t
y = 1 + t/2
z = 1 - t
where t goes from 0 to 1. Now we can substitute these expressions into the given integral and calculate the line integral as follows:
∫∫∫ zdx + xzdy + xydz
= ∫01 (1 - t)dt + ∫01 t(1 - t/2)(-dt/2) + ∫01 t(1 + t/2)(-dt)
After simplifying and evaluating the integrals, the line integral is equal to 1/2.