Evaluate the line integral c 2xyz dx + x2z dy + x2y dz where c is an oriented simple curve connecting (1, 2, 4) to (1, 4, 16).

The fact that is not described exactly is a clue to try to show the vector field is conservative. That means you need to show there is some scalar function for which ; that is, the following set of PDEs is satisfied:





Integrating both sides of the first PDE with respect to gives



Differentiating with respect to gives



Differentiating with respect to gives



So the potential scalar function is



By the fundamental theorem of calculus, we have for any oriented simple curve parameterized by a vector-valued function , starting at and terminating at ,



In this case, and , so the line integral has a value of

both graphs increase on their domain

neither graph has a y-intercept

step-by-step explanation:

the following points can be visualized easily from the graph

there is a hike in both the graphs so the graphs are increasing on their domain.x-intercept of log x and log 6x are differentthere is a vertical asymptote of both the graphs so there is no y-interceptas both the graphs are increasing so there is no horizontal asymptote