Final answer:
To compute a line integral, one needs to parametrize the function, differentiate to find the tangent vector, substitute these into the function, and integrate with respect to the parameter t over the given interval. Specific details of the parametric curve C are required to perform the actual calculation.
Step-by-step explanation:
To compute the line integral of the function f(x, y, z) = xy³z² over a given curve C represented by parameter t in the range from 0 to 1, we have to follow a step-by-step approach which can involve substituting values from the parametric equations of the curve into the function, computing the derivatives of the parametric functions, and then integrating with respect to t. The exact values of the parametric functions r(t) have been omitted in the question, but the approach remains the same.
We would obtain dr/dt, substitute into the function, and integrate over the given interval.
Without specific details of the parametric functions for the curve C, we cannot provide the explicit calculation.
However, the general idea is to parametrize the function, find the tangent vector by differentiation, and then carry out the integral on the resulting function from t=0 to t=1. It is a foundational concept in multivariable calculus, and mastering it requires practice with various kinds of curves and functions.